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Article

DP-4-Colorability on Planar Graphs Excluding 7-Cycles Adjacent to 4- or 5-Cycles

1
School of Mathematics and Science, Nanjing Tech University, Nanjing 211816, China
2
School of Mathematics & Statistics, Central China Normal University, Wuhan 430079, China
3
School of Mathematics and Computer Science & Center of Applied Mathematics, Yichun University, Yichun 336000, China
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(2), 190; https://doi.org/10.3390/math13020190
Submission received: 16 December 2024 / Revised: 4 January 2025 / Accepted: 5 January 2025 / Published: 8 January 2025
(This article belongs to the Section E: Applied Mathematics)

Abstract

:
In order to resolve Borodin’s Conjecture, DP-coloring was introduced in 2017 to extend the concept of list coloring. In previous works, it is proved that every planar graph without 7-cycles and butterflies is DP-4-colorable. And any planar graph that does not have 5-cycle adjacent to 6-cycle is DP-4-colorable. The existing research mainly focus on the forbidden adjacent cycles that guarantee the DP-4-colorability for planar graph. In this paper, we demonstrate that any planar graph G that excludes 7-cycles adjacent to k-cycles (for each k = 4 , 5 ), and does not feature a Near-bow-tie as an induced subgraph, is DP-4-colorable. This result extends the findings of the previous works mentioned above.
MSC:
05C15; 05C10

1. Introduction

The graphs discussed are simple, planar graphs. Vertex coloring stands out as a significant area of study within graph theory. A graph G with list assignment allocates a color set L ( v ) to each v V ( G ) . Considering ψ to be a proper coloring of G, which ensures that ψ ( u ) differs from ψ ( v ) across every edge u v in G. If ψ ( v ) L ( v ) in respect to every vertex v, then ψ is defined as a list coloring of G, which proposed by was Vizing et al. [1,2] independently. Considering each list assignment with a size more than k, if G has a list coloring, then we could define G as k-choosable. The least amount k is the choosable number, represented by χ l ( G ) . Thomassen [3] demonstrated the 5-choosability of planar graph, while not all planar graphs are 4-choosable, as proven by Voigt [4]. Thus, many experts focus on identifying sufficient conditions under which planar graphs have the property of 4-choosability.
In list coloring, vertex identification cannot be used due to the different lists on vertices. To address this limitation, Dvořák et al. [5] proposed DP-coloring, which enabled them to resolve a long-standing conjecture by Borodin [6], namely that every planar graph G without cycles of lengths 4 to 8 is 3-choosable. Let X , Y be two disjoint vertex sets of G, and [ X , Y ] denote the collection of edges with one end-vertex in X and the other end-vertex in Y. The following definition of DP-coloring was subsequently provided by Bernshteyn et al. [7].
Definition 1.
Consider that a graph denoted as G along with its respective list assignment L. M ( G , L ) qualifies as a cover for ( G , L ) when it meets these criteria:
  • V ( M ( G , L ) ) = w V ( G ) L w , where L w = { w } × L ( w ) = { ( w , c ) : w V ( G ) and c L ( w ) } ;
  • for each vertex w in G, the collection L w makes up a clique;
  • for every edge u v , [ L u , L v ] is a matching set (which may be empty), denoted by M u v ;
  • when u v E ( G ) , [ L u , L v ] is an empty set.
For simplicity, we denote E M = u , v V ( G ) M u v . Figure 1b,c are two distinct covers of ( C 4 , L ) with | L ( v ) | = 2 for each vertex of C 4 .
Definition 2.
Considering M ( G , L ) denoted as a cover of ( G , L ) . Then, the following two conditions are equivalent:
(i) 
M ( G , L ) includes an independent subset I with size | V ( G ) | ;
(ii) 
G has a coloring ψ on V ( G ) satisfying ( v , ψ ( v ) ) I across each vertex v. This coloring is said to M ( G , L ) -coloring.
A graph is called DP-k-colorable when it has an M ( G , L ) -coloring for every L with | L ( u ) | k and every cover M ( G , L ) . The minimum such k is the DP-chromatic number, represented as χ D P ( G ) .
For every u v E ( G ) , distinct matchings between L u and L v correspond to different covers of G. If the matching between L u and L v is { ( u , x ) ( v , x ) | x L ( u ) L ( v ) } , an M ( G , L ) -coloring of G becomes equivalent to a list coloring of G. It indicates that DP-coloring serves as a generalization of list coloring, leading to the conclusion that χ l ( G ) χ D P ( G ) . From Figure 1, H 1 has an independent set { ( v 1 , 1 ) , ( v 2 , 2 ) , ( v 3 , 1 ) , ( v 4 , 2 ) } ; hence, C 4 has an H 1 -coloring with vertices v 1 , v 3 colored 1 and vertices v 2 , v 4 colored 2, and this corresponding coloring is also a 2-list coloring of C 4 . However, C 4 does not have an H 2 -coloring for H 2 and has no 4-independent set. So, χ D P ( C 4 ) > 2 . The fact that χ D P ( C 4 ) = 3 and χ l ( C 4 ) = 2 marks an important difference between DP-coloring and list coloring.
Recently, DP-coloring has gained significant attention. Dvořák et al. [5] showed that χ D P does not exceed 5, and it can be reduced to 3 for planar graphs with a girth of at least 5. Li et al. [8] proved that any planar graphs without intersecting 5-cycles is DP-4-colorable. Li et al. [9] discussed the DP-4-colorability of planar graphs with some restrictions on cycles. For more results about DP-colorability, the readers can see in [10,11,12,13,14,15,16,17,18]. Kim and Ozeki [12] improved results about 4-choosability, any planar graph that avoids a k-cycle (where k is in the set { 3 , 4 , 5 , 6 } ) is DP-4-colorable. Moreover, Farzad [19] confirmed the 4-choosability for every planar graph lacking 7-cycles. One naturally asks whether every planar graph lacking 7-cycles has the property of DP-4-colorability. For this purpose, Kim et al. [20] demonstrated the subsequent finding.
Theorem 1.
Any planar graph that lacks both butterflies (see Figure 2b) and 7-cycles is DP-4-colorable.
On the other hand, there is another way to study the sufficient conditions for DP-4-colorability, that is, forbidden adjacent cycles. For more details, the reader can refer to [11,12,13,16,17,18,21].
Theorem 2.
Any planar graph that does not have k 1 -cycles adjacent to k 2 -cycles is DP-4-colorable, where ( k 1 , k 2 ) { ( 3 , 4 ) , ( 3 , 5 ) , ( 3 , 6 ) , ( 4 , 5 ) , ( 4 , 6 ) , ( 5 , 6 ) } .
In a planar graph G, a cluster refers to a subgraph formed by a maximal collection of 3-faces, ensuring that no other 3-face shares an edge with any face within this collection. When such a cluster contains k 3-faces, it is designated as k-cluster (see Figure 3). Adjacent cycles (faces) means that they share no less than one edge. A face f is said to be adjacent to one cluster H if there exists a face within H that shares adjacency with f. A Near-bow-tie is a graph isomorphic to the configuration depicted in Figure 2a where v is a 7-vertex. Inspired by Theorems 1 and 2, the subsequent result is formulated.
Theorem 3.
Consider a planar graph G without a Near-bow-tie as an induced subgraph. If G lacks 7-cycles that are adjacent to 4- or 5-cycles, then it is DP-4-colorable.
If a graph G is free of 7-cycles that are adjacent to either 4-cycles or 5-cycles, then G may contain 7-cycles and a butterfly. Additionally, if the graph G lacks the Near-bow-tie as an induced subgraph, then it may still contain a butterfly. Therefore, Theorem 3 generalizes Theorem 1.
A 3-cycle C is considered bad provided that it is isomorphic to the bordering of the outer face of H 7 in Figure 3; all other 3-cycles are good. Let K G denote a portion of graph G. Therefore, G is DP-k-colorable, provided that every DP-k-coloring restricted on K could be expanded to cover the whole G. To demonstrate Theorem 3, a stronger result is established as outlined below.
Theorem 4.
Consider a planar graph G without a Near-bow-tie as an induced subgraph. If G lacks 7-cycles that are adjacent to 4- or 5-cycles, then each DP-4-coloring restricted on one good 3-cycle within G can be extended to the entire graph.
At the conclusion of this section, we will present some definitions and notations. The notation [ m ] is defined as { 1 , 2 , , m } , where m is a positive integer. A k-vertex represents a node whose degree is k, while a k + (or k )-vertex represents a node whose degree is no less (or no more) than k. A face labeled as ( d 1 , d 2 , , d k ) -face corresponds to one k-face [ v 1 v 2 v k ] where v i is d i -vertex for i [ k ] . If u V ( f ) , then u is incident with f. The notation d ( C ) denotes the number of edges associated with cycle (or face) C of G.

2. Preliminaries

This section focuses on the corresponding coloring on some small subgraph, which is crucial to the proof of Theorem 4.
Lemma 1.
Consider L as a list assignment on K 3 = v 1 v 2 v 3 . If for each i [ 3 ] , the size of the list L ( v i ) is equal to 2, then K 3 has an M ( K 3 , L ) -coloring except when the subgraph of M ( K 3 , L ) induced by E M is a union of two 3-cycles.
Moreover, if L ( v i ) 2 for each i [ 3 ] and | L ( v j ) | > 2 for some j [ 3 ] , then K 3 has an M ( K 3 , L ) -coloring for any cover of ( K 3 , L ) .
Proof. 
Assume L ( v i ) = { 1 , 2 } for every index i [ 3 ] . If the subgraph of M ( K 3 , L ) generated by E M is a union of two 3-cycles, see Figure 4a, say [ ( v 1 , i ) ( v 2 , i ) ( v 3 , i ) ] for each i = 1 , 2 , then we cannot find a 3-independent set in this cover; hence, K 3 does not have an M ( K 3 , L ) -coloring for this cover. Thus, we assume that the subgraph of M ( K 3 , L ) induced by E M is not a union of two 3-cycles. Without loss of generality, suppose ( v 1 , 1 ) ( v 2 , 1 ) ( v 3 , 1 ) forms a 2-path in this cover, but ( v 1 , 1 ) ( v 3 , 1 ) E M . Then, { ( v 1 , 1 ) , ( v 3 , 1 ) , ( v 2 , 2 ) } is a 3-independent set of M ( K 3 , L ) ; see Figure 4b. Hence, we obtain an M ( K 3 , L ) -coloring of K 3 for this cover.
If there is one | L ( v i ) | 3 , for some i [ 3 ] , then we can identify a 3-independent set of M ( K 3 , L ) for any cover of ( K 3 , L ) . Consequently, an M ( K 3 , L ) -coloring of K 3 can be achieved for any cover of ( K 3 , L ) . □
Lemma 2.
Consider L as a list assignment on K 4 , where K 4 comprises two 3-faces [ x y z i ] for i { 1 , 2 } . If | L ( z i ) | 2 for each i { 1 , 2 } and | L ( x ) | , | L ( y ) | 3 , then we can assign colors to each z i using at most one color from L ( z i ) , so that K 4 has an M ( K 4 , L ) -coloring for any given cover of K 4 .
Proof. 
Assuming L ( x ) = L ( y ) = [ 3 ] and L ( z 1 ) = [ 2 ] is valid for our analysis. We assume that there is a color, say 1 L ( z 2 ) , which cannot be chosen. This means if we choose ( z 2 , 1 ) in M ( K 4 , L ) , then by Lemma 1, x , y has two available colors and the corresponding vertices of z 1 , x , y in M ( K 4 , L ) induce two triangles, say [ ( x , i ) ( y , i ) ( z 1 , i ) ] for every i { 1 , 2 } . In this case, [ ( z 2 , 1 ) ( x , 3 ) ( y , 3 ) ] is a 3-cycle of M ( K 4 , L ) . Then, for each d L ( z 2 ) { 1 } and for certain values i and j where { i , j } = [ 2 ] , the set { ( z 2 , d ) , ( x , 3 ) , ( z 1 , i ) , ( y , j ) } is an independent set in M ( K 4 , L ) . Thus, each color in L ( z 2 ) other than 1 can be chosen so that K 4 has an M ( K 4 , L ) -coloring. For vertex z 1 , we can discuss a similar method. □
Lemma 3.
Consider P = v 1 v 2 v 3 be a 2-path with a list assignment L on P, such that | L ( v i ) | 1 for every i [ 3 ] . If | L ( v 1 ) | + | L ( v 3 ) | | L ( v 2 ) | + 1 , then we can assign colors to v 1 , v 3 , such that v 2 has at least | L ( v 2 ) | 1 available colors.
Proof. 
If there is a vertex ( v 1 , a ) in M ( P , L ) such that ( v 1 , a ) is not adjacent to any vertices in L v 2 , then we have completed this section. Thus, assume each vertex in L v 1 is adjacent to a vertex of L v 2 , and the same holds for L v 3 . Given that | L ( v 1 ) | + | L ( v 3 ) | | L ( v 2 ) | + 1 , it follows that there exists a vertex in L v 2 that is adjacent to at least one vertex from both L v 1 and L v 3 . Without loss of generality, assume ( v 2 , b ) ( v 1 , a ) , ( v 2 , b ) ( v 3 , c ) E M , where a L ( v 1 ) , c L ( v 3 ) . Consequently, v 1 can be colored with color a and v 3 with c, so that v 2 has at least | L ( v 2 ) | 1 available colors. □
Lemma 4.
Let H 6 be the graph depicted in Figure 3 with a list assignment L on H 6 . If | L ( v ) | 2 for every 3-vertex of H 6 , | L ( u ) | 3 for some u { u 3 , u 5 } and | L ( v ) | 4 for other three vertices in H 6 , then we can color u with all colors but at most two of L ( u ) , ensuring that H 6 has an M ( H 6 , L ) -coloring for any given cover of ( H 6 , L ) .
Proof. 
Due to symmetry, we can assume that | L ( u 5 ) | 3 . Thus, | L ( u i ) | 4 for each i { 2 , 3 , 6 } and | L ( u i ) | 2 for each i { 1 , 4 } .
For 2-path u 1 u 5 u 4 , if | L ( u 1 ) | + | L ( u 4 ) | | L ( u 5 ) | + 1 , then by Lemma 3, we can color u 1 , u 4 such that u 5 has | L ( u 5 ) | 1 2 available colors, denoted by A ( u 5 ) . After such coloring, u 3 has at least two available colors and u i has at least three available colors for each i { 2 , 6 } . For { u 5 , u 2 , u 3 , u 6 } , by Lemma 2, we can color u 5 with at least | A ( u 5 ) | 1 = | L ( u 5 ) | 2 colors such that { u 2 , u 3 , u 6 } can be colored. This means we can color u 5 with all colors in L ( u 5 ) , except for at most two, ensuring that H 6 has an M ( H 6 , L ) -coloring for any given cover of ( H 6 , L ) , which completes this section.
Thus, | L ( u 1 ) | + | L ( u 4 ) | < | L ( u 5 ) | + 1 . Without loss of generality, we assume L ( u 1 ) = L ( u 4 ) = [ 2 ] , L ( u 5 ) = [ 4 ] . From the above discussion, we only need to prove the case that after coloring u 1 and u 4 , u 5 has exactly two available colors. Without loss of generality, we assume ( u 1 , i ) ( u 5 , i ) , ( u 4 , i ) ( u 5 , i + 2 ) E M for each i [ 2 ] . For { ( u 1 , 1 ) , ( u 4 , 1 ) } , by Lemma 2, we can color u 5 with 2 or 4 such that { u 2 , u 3 , u 6 } can be colored. For the same reason, for { ( u 1 , 2 ) , ( u 4 , 2 ) } , by Lemma 2, we can color u 5 with 1 or 3 such that { u 2 , u 3 , u 6 } can be colored. This means we can color u 5 with all colors in L ( u 5 ) , except for at most two, ensuring that H 6 has an M ( H 6 , L ) -coloring for any given cover of ( H 6 , L ) , which completes this section. □
Lemma 5.
Consider H 7 as the 7-cluster shown in Figure 3, along with a list assignment L. Let S = { u 1 , u 4 , u 5 } . If there is at most one v in S for which | L ( v ) | = 2 , while | L ( v ) | 3 for remaining vertices in S, and | L ( v ) | 4 for every vertex v in V ( H 7 ) except S, then H 7 admits an M ( H 7 , L ) -coloring for any given cover of ( H 7 , L ) .
Proof. 
By symmetry, for each i { 4 , 5 } , assume that | L ( u 1 ) | = 2 and | L ( u i ) | = 3 . Furthermore, | L ( u i ) | = 4 for each i { 2 , 3 , 6 } . Consequently, set L ( u 1 ) = { 1 , 2 } , L ( u 4 ) = L ( u 5 ) = { 1 , 2 , 3 } , L ( u 2 ) = L ( u 3 ) = L ( u 6 ) = [ 4 ] . Suppose otherwise that H 7 does not have an M ( H 7 , L ) -coloring for some cover of ( H 7 , L ) .
Suppose the subgraph induced by E M on vertices L u 1 , L u 4 and L u 5 contains two triangles in M ( H 7 , L ) since | L ( u 1 ) | = 2 ; otherwise, by Lemma 1, we can color u 1 and u 5 such that u 4 has two available colors. Then, by Lemma 2, we can color { u 2 , u 3 , u 6 , u 4 } . Therefore, this results in an M ( H 7 , L ) -coloring of H 7 , leading to a contradiction. Let [ ( u 1 , i ) ( u 4 , i ) ( u 5 , i ) ] be the two triangles for i = 1 , 2 and ( u 4 , 3 ) ( u 5 , 3 ) E M .
Since H 7 does not have an M ( H 7 , L ) -coloring, we have the following proposition, Proposition 1, constructed easily using Lemma 1. □
Proposition 1.
If u 1 , u 4 , u 5 are colored, then the subgraph of M ( H 7 , L ) induced by edge E M between { { u i } × A ( u i ) | i = 2 , 3 , 6 } is two triangles, where A ( u i ) denotes the available colors in L ( u i ) after coloring { u 1 , u 4 , u 5 } .
By Proposition 1 and Lemma 1, there are four triangles between { u i } × L ( u i ) ( i { 2 , 3 , 6 } ) in M ( H 7 , L ) . Let [ ( u 2 , i ) ( u 3 , i ) ( u 6 , i ) ] be the four triangles in M ( H 7 , L ) for each i [ 4 ] . We assume that ( u 1 , 1 ) ( u 2 , 1 ) E M . Assume ( u 1 , 1 ) ( u 3 , i ) E M for some i [ 4 ] . By symmetry, we only need to discuss two cases: i = 1 or i = 2 .
If i = 2 , then by Proposition 1, ( u 4 , 3 ) ( u 3 , 1 ) E M for 3-independent set { ( u 1 , 1 ) , ( u 4 , 3 ) , ( u 5 , 2 ) } and ( u 4 , 2 ) ( u 3 , 1 ) E M for 3-independent set { ( u 1 , 1 ) , ( u 4 , 2 ) , ( u 5 , 3 ) } ; contrary to this, [ L u 3 , L u 4 ] is a matching set.
If i = 1 , then by Proposition 1, ( u 5 , 2 ) ( u 2 , j ) , ( u 4 , 3 ) ( u 3 , j ) E M for some j { 2 , 3 , 4 } , say j = 2 . In this case, for 3-independent set { ( u 1 , 2 ) , ( u 4 , 3 ) , ( u 5 , 1 ) } , ( u 1 , 2 ) ( u 2 , 2 ) , ( u 1 , 2 ) ( u 3 , k ) E M for some k { 3 , 4 } . The independent set { ( u 1 , 2 ) , ( u 4 , 1 ) , ( u 5 , 3 ) } , ( u 4 , 1 ) ( u 3 , 2 ) E M , contrary to [ L u 3 , L u 4 ] , is a matching set.
Lemma 6.
Consider H 7 as the 7-cluster shown in Figure 3, along with a list assignment L. If | L ( u i ) | 3 for i = 4 , 5 and | L ( v ) | 4 for the other vertices of H 7 , then we can color u 1 with all colors but at most one in L ( u 1 ) , ensuring that H 7 maintains an M ( H 7 , L ) -coloring for any given cover of ( H 7 , L ) .
Proof. 
Without loss of generality, we assume that for each i { 4 , 5 } , L ( u i ) = [ 3 ] and L ( u j ) = [ 4 ] for each j { 1 , 2 , 3 , 6 } . For any given cover of ( H 7 , L ) , by contradiction, we assume that there are two colors that cannot be chosen in L ( u 1 ) , say { 1 , 2 } . Since [ 2 ] L ( u 1 ) , by Lemma 5, there is an M ( H 7 , L ) -coloring for any given cover of ( H 7 , L ) , a contradiction. This means we can color u 1 with all colors, but only use at most one in L ( u 1 ) , ensuring that H 7 maintains an M ( H 7 , L ) -coloring for any cover of ( H 7 , L ) . □
The following definition is due to Sittitrai and Nakprasit [22].
Definition 3.
Let M ( G , L ) represent a cover of ( G , L ) , with G as an induced subgraph within G. The following definitions apply:
(i) 
A list assignment L | G is considered a restriction of L on G if L | G ( u ) = L ( u ) for every vertex of G .
(ii) 
A graph M ( G , L ) | G is termed a restriction of M ( G , L ) on G if M ( G , L ) | G = M ( G , L ) [ { v × L ( v ) : v V ( G ) } ] . According to the cover definition, M ( G , L ) | G M ( G , L | G ) .
Assuming that G has an M ( G , L | G ) -coloring with an independent set I , the residual list assignment L * for G G is defined as follows:
L * ( y ) = L ( y ) x y E ( G ) { c L ( y ) : ( x , c ) ( y , c ) E ( H ) and ( x , c ) I }
for every y V ( G G ) . The residual cover M ( G G , L * ) * is established by M ( G , L ) [ { w × L * ( w ) : w V ( G G ) } ] .
Lemma 7.
([22]). Let M ( G , L ) represent a cover of ( G , L ) with G as an induced subgraph of G. If G maintains an M ( G , L | G ) - coloring, then the residual cover M ( G G , L * ) * serves as a cover for ( G G , L * ) , where L * represents the residual list assignment. Additionally, if G G is colorable with respect to M ( G G , L * ) * , then G is also M ( G , L ) -colorable.
For the remainder of this paper, G is regarded as an induced subgraph of ( G , L ) , where | L ( v ) | 4 for every vertex v of G. By Lemma 7, to establish that G is M ( G , L ) -colorable, it suffices to demonstrate that G maintains an M ( G , L | G ) -coloring and G * maintains an M ( G G , L * ) * -coloring.

3. Reducible Configurations

Consider ( G , C 0 ) as a counterexample to Theorem 4, where the number of vertices is minimized, and within that constraint, the number of edges is also minimized, with C 0 representing a good 3-cycle. The proof presented in [12] establishes that any planar graph devoid of 3-cycles is DP-4-colorable. Therefore, it is reasonable to assume that G includes a 3-cycle. Indeed, we can always identify a good 3-cycle within G by Lemma 5. Let L represent a 4-list assignment on the vertex set V ( G ) , and C 0 as the outer cycle of G. Let M ( G , L ) be any cover of ( G , L ) and ϕ 0 the M ( G , L ) -coloring restricted on C 0 . The following lemma holds true when G lacks any 7-cycle that is adjacent to 4- or 5-cycles.
Lemma 8.
G contains no k-cluster for k 8 ; moreover, the graphs of Figure 3 are all k-clusters of G, where k { 4 , 5 , 6 , 7 } .
Consider C representing a cycle within G. The notation i n t ( C ) (or e x t ( C ) ) is employed to indicate the collections of vertices lying inside (or outside) the cycle C. A cycle C is referred to as separating if it divides the vertex set such that both i n t ( C ) and e x t ( C ) include vertices. A vertex u in G is classified as internal if u V ( C 0 ) . Additionally, a face f is classified as internal if all vertex of f is internal.
Lemma 9.
The statements below are all valid:
(i) 
V ( G ) V ( C 0 ) ;
(ii) 
G does not have any good 3-cycle that is separating;
(iii) 
the degree of every internal vertex in G is at least 4.
Proof. 
(i) Assume, for the purpose of contradiction, that V ( G ) = V ( C 0 ) . Since C 0 is a good 3-cycle, the M ( G , L ) -coloring restricted on C 0 would also serve as an M ( G , L ) -coloring for G, which contradicts the initial assumption.
(ii) Suppose C is a separating good 3-cycle. Given the property of minimality in G, we can extend ϕ 0 to ext(C) C , say ϕ 1 . This extension can then be restricted to C and further extended to create an M ( G , L ) -coloring ϕ 2 for i n t ( C ) . Consequently, by merging ϕ 1 and ϕ 2 , we obtain an M ( G , L ) -coloring for G, contrary to the assumption.
(iii) Let v represent an internal 3 -vertex. Given that G is minimal, the M ( G , L ) -coloring restricted on C 0 can be applied to G v . Since d G ( v ) 3 , v has at least one available color for any cover of ( G , L ) . This means that G has an M ( G , L ) -coloring, a contradiction. □
Lemma 10.
No two internal ( 4 , 4 , 4 ) -faces in G can share a common edge.
Proof. 
Assume, for the sake of contradiction, that f i = [ x y z i ] ( i = 1 , 2 ) are two internal (4, 4, 4)-faces. Denote G = G f 1 f 2 . By the minimality of G, the graph G maintains an M ( G , L | G ) -coloring extended by ϕ 0 on C 0 , and L * , M ( G G , L * ) * are the residual list assignment and the residual cover of ( f 1 f 2 , L * ) . Since each vertex of f 1 f 2 is a 4-vertex, | L * ( z i ) | 2 for every i { 1 , 2 } and | L * ( x ) | , | L * ( y ) | 3 . By Lemmas 2 and 7, we can obtain an M ( G , L ) -coloring for G, which contradicts the original assumption. □
A cluster is special if it is one of { H 4 4 , H 5 2 , H 6 , H 7 } in Figure 3 with an internal (4, 4, 4)-face [ u 2 u 3 u 6 ] (or [ u 2 u 5 u 6 ] by symmetry if the cluster is H 6 ); otherwise non-special. An internal 4 + -vertex v located inside a cluster H is classified as i-type with respect to H if it is connected to precisely i edges of H. Moreover, if H is a special cluster, then this 4 + -vertex v is referred to as i-type good; otherwise, it is considered i-type bad. The following result is obvious from the fact that G contains no 7-cycle adjacent to a 4- or 5-cycle.
Lemma 11.
(i) 
A 1-cluster is adjacent to no more than a 4-face.
(ii) 
The face of G adjacent to i-cluster should be 8 + -face for each i { 4 , 5 , 6 , 7 } .
(iii) 
The face of G adjacent to 2- or 3-cluster should be 4-face or 8 + -face; moreover, s 2-cluster is adjacent to no more than two non-adjacent 4-faces, where these two 4-faces are adjacent to different 3-face of this 2-cluster. Similarly, a 3-cluster is adjacent to no more than one 4-face, which is incident to one 4-type 4-vertex to this 3-cluster.
Lemma 12.
No internal 5-vertex of G can belong to two clusters which are special.
Proof. 
Let v represent an internal 5-vertex, and let its neighborhood be N ( v ) = { v i : i [ 5 ] } . Assume, for the purpose of contradiction, that v is a vertex of two special clusters K 1 , K 2 . Without loss of generality, assume the internal (4, 4, 4)-faces of K 1 and K 2 are [ v 1 v 2 v 12 ] and [ v 3 v 4 v 34 ] , respectively. Define H 1 and H 2 as the graphs induced by vertex set { v 1 , v 2 , v 12 , v } and { v 3 , v 4 , v 34 , v } , respectively. Clearly, H i is isomorphic to K 4 . Let G = G H 1 H 2 and L * , M ( G G , L * ) * denote the residual list assignment and the residual cover of G G . Due to the minimality property of ( G , C 0 ) , G maintains an M ( G , L | G ) -coloring extended from ϕ 0 . Notably, | L * ( v ) | and | L * ( v i ) | are both at least 3 for each i [ 4 ] , while | L * ( v 12 ) | , | L * ( v 34 ) | 2 . Applying Lemma 2 to H 1 , we can color v with at least two colors in L * ( v ) , ensuring that H 1 maintains an M ( H 1 , L * | H 1 ) -coloring for every cover of ( H 1 , L * | H 1 ) . Applying Lemma 2 to H 2 subsequently yields an M ( G G , L * ) * -coloring for G G . By Lemma 7, this implies that G maintains an M ( G , L ) -coloring, contrary to the assumption. □
Internal 6-vertex v is classified as special if it qualifies as a 4-type good vertex to an internal 6 + -cluster K 1 and as a 2-type good vertex in association with a k 2 -cluster K 2 , where 4 k 2 5 (see Figure 5a). Internal 6-vertex v is poor if it is a 3-type good vertex to an internal 6-cluster K 1 and a 3-type good vertex to a k 2 -cluster K 2 , where 5 k 2 6 (see Figure 5b).
Lemma 13.
Consider v as either a special or poor 6-vertex contained in K 1 and K 2 (see Figure 5a,b). Let G = G K 1 , with L * , M ( G G , L * ) * representing the residual list assignment and residual cover of G G . Then, | L * ( v ) | 3 if v is a special 6-vertex, and | L * ( v ) | 2 if v is a poor 6-vertex.
Proof. 
Since v is either a special or poor 6-vertex, K 2 is a special cluster. Denote [ x y z ] to represent the (4, 4, 4)-face of K 2 . Denote G = G { x , y , z } = G K 1 { x , y , z } , with L * * , M ( G G , L * * ) * * as the residual list assignment and residual cover for G G . This implies that | L * * ( x ) | 2 and | L * * ( u ) | 3 for each u { y , z } .
If v is a special 6-vertex, then | L * * ( v ) | 4 . If v is a poor 6-vertex, then | L * * ( v ) | 3 . In either case, by Lemma 2, we can precolor v using all colors in L * * ( v ) except for at most one, allowing the vertices { x , y , z } to be colored as well. This means | L * ( v ) | 3 when the vertex v is classified as a special 6-vertex and | L * ( v ) | 2 when v is a poor 6-vertex. □
We use a recursive method to define the poor 7-vertex, the special 7-vertex, and the special 8-vertex.
Definition 4.
Consider v as a 7-vertex of G, which is 4-type good to one internal 6-cluster K 1 and 3-type good to one internal 6-cluster K 2 .
(i) 
v is a  poor 7 0 -vertex  if K 1 contains two 3-type vertices, each of which is in {3-type good 5-vertex, 3-type poor 6-vertex};
(ii) 
v is a  poor 7 m -vertex  if K 1 contains one 3-type poor 7 m 1 -vertex and one 3-type vertices which is in {3-type good 5-vertex, 3-type poor 6-vertex, 3-type poor 7 k -vertex ( k m 1 ) }.
Then, v is called a  3-type poor 7-vertex  if v is a 3-type poor 7 m -vertex, where m N .
Let A 1 = { 3-type good 5-vertex, 3-type poor 6-vertex, 3-type poor 7-vertex}.
Definition 5.
Consider v as a 7-vertex of G which is 4-type good to one internal 6 + -cluster K 1 and 3-type good to one internal 6-cluster K 2 .
(i) 
v is a  special 7 0 -vertex  if K 2 contains one vertex of A 1 and one vertex of {4-type good 5-vertex, 4-type special 6-vertex};
(ii) 
v is a  special 7 m -vertex  if K 2 contains one vertex of A 1 and one special 7 m 1 -vertex.
Then, v is called a  4-type special 7-vertex  if v is a 4-type special 7 m -vertex, where m N .
Let A 2 = { 4-type good 5-vertex, 4-type special 6-vertex, 4-type special 7-vertex}.
Definition 6.
Consider v as an 8-vertex of G which is 4-type good to one internal 7-cluster K 1 and 4-type good to one internal 6 + -cluster K 2 .
(i) 
v is a  special 8 0 -vertex  if K 1 contains two vertices which are in A 2 ;
(ii) 
v is a  special 8 m -vertex  if K 1 contains one special 8 m 1 -vertex and one vertex of A 2 { 4-type special 8 l -vertex ( l m 1 ) } .
Then, v is called a  special 8-vertex  if v is a special 8 m -vertex, where m N .
By Definition 6, in order to distinguish the above clusters K 1 and K 2 , K 1 is called type I and K 2 is called type II.
Lemma 14.
Consider v as a 3-type vertex to an internal cluster K. Let G = G K , with L * and M ( G G , L * ) * as the residual list assignment and residual cover on G G , respectively. If v A 1 , then | L * ( v ) | 2 .
Proof. 
If v is a 5-vertex, then the cardinality of L * ( v ) is at least 2. If v is a 3-type poor 6-vertex, then | L * ( v ) | 2 by Lemma 13. Then, v is a poor 7-vertex. By Definition 4, assuming v is a poor 7 m -vertex, m N . We use the symbol shown in Figure 5c, where K 2 = K and K 1 are 6-clusters. Let [ x y z ] and [ x y z ] be (4, 4, 4)-faces in K 1 and K 2 , respectively. Since v is poor, K 1 is also internal, and u , w A 1 . Let G = G K 1 = G ( K 1 K 2 ) , and let L * * and M ( G G , L * * ) * * denote the residual list assignment and residual cover of G G . We use the induction hypothesis on m. If m = 0 , then { u , w } is a 3-type 5- or poor 6-vertex, by the above discussion, | L * * ( u ) | , | L * * ( w ) | 2 . Clearly, | L * * ( i ) | = 4 for each i { x , y , z , v } . Lemma 4 provides a minimum of two colors available in L * * ( v ) to color K 1 . This means | L * ( v ) | 2 . By the induction hypothesis, it is assumed that the lemma is valid for m n 1 .
Next, we prove that the lemma is valid if m = n .
By Definition 4, each of { u , w } is a 5-, poor 6-, or 7 l -vertex ( l n 1 ) , using Lemma 13 and the induction hypothesis, | L * * ( u ) | 2 and | L * * ( w ) | 2 . Then, using Lemma 4, we have completed this section. □
Lemma 15.
Consider v as a 4-type vertex to an internal cluster K. Let G = G K with L * and M ( G G , L * ) * representing the residual list assignment and residual cover of G G .
(i) 
If v A 2 , then the cardinality of L * ( v ) is at least 3;
(ii) 
If v is a special 8-vertex and K is type II, then | L * ( v ) | 3 .
Proof. 
(i) Clearly, | L * ( v ) | 3 if v is a 5-vertex. If v is a 4-type special 6-vertex, then | L * ( v ) | 3 by Lemma 13. Then, v is a special 7-vertex. By Definition 5, assuming v is a special 7 m -vertex, m N . We use the labeling shown in Figure 5d, where K 1 = K and K 2 is an internal 6-cluster. Then, the (4, 4, 4)-face corresponding to the cluster K 2 is denoted by [ x y z ] and u A 1 . Let G = G K 2 = G ( K 1 K 2 ) , with L * * and M ( G G , L * * ) * * representing the residual list assignment and residual cover of G G . Then, for each vertex q { v , x , y , z } , | L * * ( q ) | = 4 . Since u A A 1 , by Lemma 14, | L * * ( u ) | 2 .
Next, we use the induction hypothesis on m.
If m = 0 , then w is a good 5- or special 6-vertex. According to Lemma 13, we have | L * * ( w ) | 3 . Without loss of generality, we can take L * * ( w ) = [ 3 ] , L * * ( u ) = [ 2 ] , L * * ( q ) = [ 4 ] for each q { x , y , z , v } . By Lemma 4, for { 3 , 4 } L ( v ) , there is an M ( K 2 , L * * ) -coloring, say ϕ 1 , with ϕ 1 ( v ) = 3 . Similarly, for { 1 , 2 } L ( v ) , there is an M ( K 2 , L * * ) -coloring, say ϕ 2 , with ϕ 2 ( v ) = 1 ; for { 2 , 4 } L ( v ) , there is an M ( K 2 , L * * ) -coloring, say ϕ 3 , with ϕ 3 ( v ) = 2 . Hence, | L * ( v ) | 3 .
By the induction hypothesis, it is assumed that the lemma is valid for m n 1 .
Next we prove that the lemma is valid if m = n . By Definition 5, w is a special 7 n 1 -vertex. By the induction hypothesis, | L * * ( w ) | 3 . By Lemma 4 and the similar reasoning for the case m = 0 , we have | L * ( v ) | 3 .
(ii) Since v is a special 8-vertex, v is a special 8 m -vertex, m N . We use the induction hypothesis on m. If m = 0 , then v is a special 8 0 -vertex, K is an internal 6 + -cluster and v is also a vertex of internal 7-cluster K 1 , and K 1 contains two vertices in A 2 , say u , w . Let G = G K 1 = G ( K K 1 ) , with L * * and M ( G G , L * * ) * * representing the residual list assignment and residual cover of G G . Let [ x y z ] be the (4, 4, 4)-face in K 1 . Then, for each vertex q { v , x , y , z } , | L * * ( q ) | = 4 . Since u , w A 2 , by Lemma 15(i), | L * * ( u ) | 3 , | L * * ( w ) | 3 . By Lemma 6, we can assign colors to v using all but at most one color, ensuring that K 1 maintains an M ( K 1 , L * * ) -coloring for any given cover of ( K 1 , L * * ) . This means | L * ( v ) | 4 1 3 .
By the induction hypothesis, it is assumed that the lemma is valid for m n 1 .
Next, we prove that the lemma is valid if m = n . By Definition 6, assuming that u is a special 8 n 1 -vertex and w is a vertex from A 2 { the 4-type special 8 l -vertex ( l n 1 ) } . By the induction hypothesis, the cardinality of both L * * ( u ) and L * * ( w ) is at least 3. By Lemma 6, we have | L * ( v ) | 3 . □
Lemma 16.
If v is a poor 7-vertex in G, then it cannot be a special 7-vertex of G.
Proof. 
Since v is a poor 7-vertex, we will consider v as a good 4-type vertex in relation to the internal 6-cluster K 1 and as a good 3-type vertex to internal 6-cluster K 2 . Clearly, v A 1 . Suppose, on the contrary, that v is also a special 7-vertex. This means v A 2 . Let G = G K 1 K 2 , with L * and M ( G G , L * ) * representing the residual list assignment and residual cover for G G . Given the minimal property of G, G has an M ( G , L | G ) -coloring that extends ϕ 0 . Since v A 2 , there are three available colors in L * ( v ) to choose, so that K 2 can be colored by Lemma 15. Since v A 1 , there are two available colors in L * ( v ) to choose, so that K 1 can be colored by Lemma 14. Since | L * ( v ) | = 4 , we can precolor v so that K 1 K 2 can be colored. By Lemma 7, G is ( M , L ) -coloring for any given cover of ( G , L ) , leading to a contradiction. □
Lemma 17.
Consider v as a 7-vertex of G, where it is 4-type to a 7-cluster K 1 and 3-type to a 6 -cluster K 2 . Denote [ x , y , z ] represent a (4, 4, 4)-face within K 1 and u , w V ( K 1 ) . If u , w A 2 { the special 8-vertex and K 1 is type II}, then K 2 cannot be special.
Proof. 
Assume, to the contrary, that K 2 is special. In this case, let [ x y z ] represent the internal (4, 4, 4)-face within K 2 with y , z N ( v ) . Since u , w A 2 { the special 8-vertex and K 1 is type II}, K 1 is internal. Define G = G K 1 { x , y , z } , and let L * and M ( G G , L * ) * be the residual list assignment and the residual cover of G G . In this setup, | L * ( x ) | 2 and | L * ( y ) | , | L * ( z ) | , | L * ( v ) | 3 . By Lemma 15, | L * ( u ) | , | L * ( w ) | 3 . Clearly, | L * ( x ) | , | L * ( y ) | and | L * ( z ) | 4 . According to Lemma 2, v has a minimum of two colors that can be used from L * ( v ) , allowing for the coloring of the set { x , y , z } . Then, by Lemma 5, K 1 has an M ( K 1 , L * | K 1 ) * -coloring for any cover of ( K 1 , L * ) . Therefore, G G maintains an M ( G G , L * ) * -coloring. By the minimality of G, G also maintains an M ( G , L | G ) -coloring. By Lemma 7, G maintains an M ( G , L ) -coloring, a contradiction. □
Lemma 18.
Consider H 6 G as an internal special 6-cluster with a (4, 4, 4)-face [ u 2 u 3 u 6 ] , shown in Figure 3. If u i A A 1 for each i { 1 , 4 } , then d ( u 5 ) 6 and u 5 A 2 { the special 8-vertex, and H 6 is type II}.
Proof. 
By Lemma 10, d ( u 5 ) 5 . Denote G = G H 6 with L * and M ( H 6 , L * ) * as the residual list assignment and residual cover of G G . By the minimal property of G, G maintains an M ( G , L | G ) -coloring that extends ϕ 0 . Suppose otherwise that u 5 A 2 or u 5 is a special 8-vertex with H 6 type II. Then, by Lemma 15, | L * ( u 5 ) | 3 . Since u i A 1 for each i { 1 , 4 } , by Lemma 14, | L * ( u i ) | 2 . Note that | L * ( u i ) | 4 for each i { 2 , 3 , 6 } . By Lemma 4, H 6 has an M ( H 6 , L * ) * -coloring. According to Lemma 7, G admits an M ( G , L ) -coloring, which is a contradiction. □
Lemma 19.
Consider H 7 G as an internal 7-cluster (see Figure 3) and X = { u 1 , u 4 , u 5 } .
(i) 
If X contains two vertices in A 2 { the special 8-vertex, and H 7 is type II}, then the third vertex of X is a 7 + -vertex;
(ii) 
X does not contain three vertices in A 2 { the special 8-vertex, and H 7 is type II}.
Proof. 
(i) By symmetry, assume u 4 , u 5 A 2 { the special 8-vertex, and H 7 is type II}. Suppose, to the contrary, that d ( u 1 ) 6 .
Define G = G H 7 with L * and M ( H 7 , L * ) * as the residual list assignment and the residual cover of G G . By the minimal property of G, G maintains an M ( G , L | G ) -coloring that extends ϕ 0 , with | L * ( u 1 ) | 2 . Note that | L * ( u i ) | 4 for each i { 2 , 3 , 6 } . Since u 4 , u 5 A 2 { the special 8-vertex, and H 7 is type I}, by Lemma 15, | L * ( u i ) | 3 , for each i { 4 , 5 } . By Lemma 5, an M ( H 7 , L * ) * -coloring of H 7 can be achieved. Then, by Lemma 7, G admits an M ( G , L ) -coloring, a contradiction.
(ii) follows easily by Lemmas 5 and 15. □

4. Proof of Theorem 4

Consider V as the family of vertices of G and F as the family of faces. An initial charge is defined as μ ( x ) = d ( x ) 4 for every x V F { C 0 } , while μ ( C 0 ) = d ( C 0 ) + 4 , where C 0 is designated as the outer face of G. According to Euler’s Formula, the sum x V F μ ( x ) = 0 . Let μ * ( x ) represent the charge of x V F after the discharging procedure. To reach a contradiction, the purpose of this is to demonstrate that μ * ( x ) 0 for all x V F { C 0 } and that μ * ( C 0 ) > 0 .
The discharging rules are as follows:
(R1) 
Every 5- or 8 + -face distributes 1 2 to every neighboring 3-face, while a k-face ( k = 6 , 7 ) distributes k 4 k to every neighboring 3-face;
Consider v as a vertex of k-face f and let t represent the number of edges of f that are incident to v, such that these edges are not in any 3-faces. In cases where k 8 , f distributes t 4 to v; f distributes t ( k 4 ) 2 k to v when k { 6 , 7 } ;
(R2) 
Every 2-cluster obtains 1 4 from its 3-type 4 + -vertex and 1 4 from its 2-type 4 + -vertex, which is on a 4-face adjacent to this cluster;
(R3) 
Every 3-cluster which is adjacent to 4-face obtains 1 2 from its 3-type 4 + -vertex and 1 4 from its 2-type 4 + -vertex; otherwise, 3-cluster obtains 1 4 from its 3-type 4 + -vertex;
(R4) 
Every 4-cluster obtains the charge from its incident vertex v, defined by
1 2 , if v is a 2 - type good 5 + - vertex or a 3 - type 4 + - vertex ; 1 , if v is a 5 - type 5 + - vertex ; 3 2 , if v is a 4 - type 5 + - vertex .
(R5) 
Every 5-cluster obtains the charge from its incident vertex v defined by
1 2 , if v is a 2 - type good 5 + - vertex or 3 - type 4 - vertex or 3 - type bad 5 - , 6 - , 7 - vertex ; 1 , if v is a 3 - type good 5 - , 6 - , 7 - vertex or 3 - type 8 + - vertex or 5 - type 5 - vertex ; 3 2 , if v is a 4 - type 5 + - vertex .
(R6) 
Every 6-cluster K obtains the charge from its incident vertex v defined by
1 2 , if v is a 3 - type 4 - vertex or 3 - type bad 5 - , 6 - , or 7 - vertex ; 1 , if v A 1 or v is a 3 - type good , non - poor 6 - or 7 - vertex , while K is not internal ; 2 , if v is a 4 - type good 6 + - vertex , while K is internal and contains two vertices in A 1 , or a 4 - type good 9 + - vertex ; 3 2 , if v is a 4 - type 5 + - vertex except the above 4 - type vertices , or 3 - type 8 + - vertex , or 3 - type good , non - poor 6 or 7 - vertex , while K is internal .
(R7) 
Every 7-cluster K obtains the charge from its incident internal vertex v defined by
3 2 , if v A 2 { the special 8 - vertex and K is type II } , or v is a 7 , 8 - vertex or non - special 6 - vertex while K is not internal , 2 , if v is a non - special 6 - vertex while K is internal , or a non - special 7 , 8 - vertex while K is internal and contains no two vertices in A 2 { the special 8 - vertex and K is type II } , 5 2 , if v is a non - special 7 , 8 - vertex while K is internal and contains two vertices ( except v ) in A 2 { the special 8 - vertex and K is type II } or 9 + - vertex .
(R8) 
C 0 receives μ ( v ) from every incident vertex, and distributes 1 to every non-internal 3-face;
(R9) 
At last, each face excepting C 0 transfers its present charge to C 0 .
The following lemma can be easily obtained by discharging rules.
Lemma 20.
Assuming u is a k-type vertex within a cluster, if d G ( u ) 4 , then u gives at most
1 2 , if k = 2 , 3 2 , if k = 3 , 5 2 , if k = 4 , 1 , if k = 5 . to this cluster .
We begin by examining the final charge assigned to each vertex of G. Consider v as a vertex of G. If v V ( C 0 ) , then according to rule (R8), we have μ * ( v ) 0 . Now, consider the situation in which v is not in V ( C 0 ) . By applying Lemma 9 (iii), the minimal degree of G is 4.
  • d ( v ) = 4 . Suppose v is a k-type vertex of cluster K for some k { 2 , 3 , 4 } . If k = 4 , then v does not participate in the discharge process, hence μ * ( v ) = μ ( v ) = 0 . If k = 3 , then K is a 2 + -cluster and v must be contained in at least one 8 + -face. By (R2)–(R6), v gives 1 4 or 1 2 to K. By (R1), v obtains at least either 1 4 if v is incident with a 4-face or 1 2 otherwise. Therefore, if v is not incident with 4-face, then μ * ( v ) is not negative. Assuming v is incident to a single 4-face. It can be inferred from Lemma 11 that K is a 2-cluster. By (R2) and (R1), μ * ( v ) = μ ( v ) + 1 4 1 4 = 0 . If k = 2 , then K is a 1 + -cluster. By (R2)–(R5), v gives a charge to K when K is a 2- or 3-cluster. If K is a 2- or 3-cluster, then v gives 1 4 to K when v is incident with a 4-face adjacent to K according to (R2)–(R3). Given that G lacks 7-cycles that are adjacent to 4- or 5-cycles, and according to Lemma 11, the three other faces incident with v in clockwise direction are 4-, 6 + -, and 8 + -faces, respectively. By applying (R1), we have μ * ( v ) = μ ( v ) 1 4 + 1 6 + 1 4 > 0 .
  • d ( v ) = 5 . Assume that v represents a k-type vertex of cluster K for some k { 2 , 3 , 4 , 5 } . By applying Lemma 20, it can be inferred that μ * ( v ) is at least μ ( v ) 1 , which yields μ * ( v ) 0 when k = 5 . If k = 4 , then K is a 3 + -cluster and v is contained in two 8 + -faces. By (R3)–(R7), v gives at most 3 2 to K. By (R1), v receives 1 4 from each incident 8 + -face, yielding a total of 1 2 . Therefore, we find that μ * ( v ) μ ( v ) + 1 2 3 2 = 0 . If k = 3 , then K is a 2 + -cluster. By rules (R2)–(R6), v contributes no more than 1 to K for d ( v ) = 5 . If v is part of only a single cluster, then it follows that μ * ( v ) = μ ( v ) 1 = 0 . Thus, let us consider the scenario where v is a 2-type vertex of other cluster, referred to as L. By Lemma 11, v is contained in two 8 + -faces. According to rules (R2)–(R5), v contributes 1 2 to L when L is a special 4-, 5-cluster. By Lemma 12, K is not a special cluster. According to rules (R2)–(R6), v contributes no more than 1 2 to K. Then, μ * ( v ) = μ ( v ) 1 2 × 2 = 0 . The case of k = 2 can be proved similarly as k = 3 .
  • d ( v ) = 6 . Assume v is contained in no more than three clusters. By (R2)–(R7), v contributes no more than 2 to its incident cluster. Then, μ * ( v ) μ ( v ) 2 = 0 when v is contained in only one cluster. If v is contained in three clusters, then it is a 2-type vertex for each cluster. According to Lemma 20, μ * ( v ) = μ ( v ) 1 2 × 3 > 0 . Therefore, we consider the case where v belongs to exactly two clusters, say K 1 , K 2 . Suppose firstly that v is a 2-type vertex for K 1 . According to Lemma 20, assume v is a 4-type vertex to K 2 and v gives more than 3 2 to K 2 . By (R6)–(R7), v gives 2 to K 2 , where K 2 is an internal 6- or 7-cluster. By (R6) and (R7), either v is a good but not special 6-vertex, or K 2 is a special 6-cluster containing two vertices in A 1 . In the latter situation, by Lemma 18, v is also good but not a special 6-vertex. In either case, v is a non-special 6-vertex, while K 2 is internal and special. By the definition of the special 6-vertex, K 1 is not a special 4-, 5-cluster. By (R2)–(R5), v gives no charge to K 1 . In this case, μ * ( v ) = μ ( v ) 2 = 0 . Now, let us consider the scenario where v acts as a 3-type vertex for K 1 . Due to the symmetric properties, v is necessarily classified as a 3-type vertex in relation to K 2 . According to rules (R2)–(R6), we can assume that K i is 5 + -cluster for each i { 1 , 2 } . By (R5)–(R6), assume v gives 3 2 to K 1 . Then, K 1 is an internal 6-cluster and v is a 3-type good 6-vertex which is not poor. By the definition of a poor 6-vertex, K 2 is not a special 5- or 6-cluster. Hence, v is a 3-type bad 6-vertex to K 2 . By (R5)–(R6), v contributes at most 1 2 to K 2 . Thus, we have μ * ( v ) = μ ( v ) 3 2 1 2 = 0 .
  • d ( v ) = 7 . Then, v is contained in no more than three clusters and μ ( v ) = 3 . By Lemma 20, we only need to discuss the case that v is contained in two clusters, say K 1 and K 2 , and v is classified as a 4-type vertex to K 1 and a 3-type vertex to K 2 . Since v gives at most 3 2 to K 2 , we need only show the case that v gives at least 2 to K 1 cannot occurs by (R2)–(R7). Further, by rules (R2)–(R6), v contributes at most 5 2 to K 1 . If v contributes 5 2 to K 1 , then K 1 is an internal 7-cluster and contains two vertices in A 2 { the special 8-vertex and K 1 is type II} by (R7). By Lemma 17, K 2 is not special. According to (R2)–(R6), v contributes at most 1 2 to K 2 . Hence, we find that μ * ( v ) μ ( v ) 5 2 1 2 = 0 , and we complete this section. If v gives 2 to K 1 , then one of the following applies: (i) v is a non-special 7-vertex, while K 1 is classified as an internal 7-cluster, lacking two vertices of A 2 { the special 8-vertex and K 1 is type II} by (R7); (ii) K 1 is an internal special 6-cluster that includes two vertices from A 1 by (R6). When (i) occurs, K 2 is not a 6-cluster for G and contains no Near-bow-tie flaw. Following rules (R2)–(R5), v contributes no more than 1 to K 2 . Therefore, we have μ * ( v ) μ ( v ) 2 1 = 0 , concluding our argument. When (ii) occurs, v is poor if K 2 is an internal and special 6-cluster. By the same rules (R2)–(R6), v contributes no more than 1 to K 2 , again leading to μ * ( v ) μ ( v ) 2 1 = 0 , which completes the proof.
  • d ( v ) = 8 . Hence, μ ( v ) = 4 . By applying Lemma 20, suppose v belongs to at least two clusters. If v is present in four clusters, then μ * ( v ) 4 4 × 1 2 > 0 according to Lemma 20. For the case where v belongs to three clusters, we have μ * ( v ) 4 2 × 3 2 1 2 > 0 or μ * ( v ) 4 5 2 2 × 1 2 > 0 by Lemma 20. Thus, suppose v is contained in just two clusters, say K 1 and K 2 . By Lemma 20, suppose v gives 5 2 to K 1 and v is a 4-type vertex of K 1 . According to (R3)–(R7), K 1 is an internal 7-cluster and contains two vertices in A 2 { the special 8-vertex and K 1 is type II}. We only need to prove cases that v gives 2 or 5 2 to K 2 cannot occur. Suppose otherwise, by (R2)–(R7), one of the following applies: (i) K 2 is an internal and special 6-cluster and contains two vertices in A 1 ; (ii) K 2 is an internal 7-cluster and v is a non-special 8-vertex. By Definition 6, v would indeed be a special 8-vertex if K 2 is an internal 7-cluster, contradicting the assumption that v is a non-special 8-vertex. This makes the second case (ii) impossible. Therefore, K 2 is an internal special 6-cluster containing two vertices from A 1 . Let G = G K 1 K 2 , where L * , M ( G G , L * ) * be the residual list assignment and the residual cover of G G = K 1 K 2 . By the minimal property of G, G admits M ( G , L | G ) -coloring extending ϕ 0 with | L * ( v ) | 4 . By Lemmas 4 and 14, v has at least two available colors to color K 2 . Thus, by Lemmas 5 and 15, K 1 K 2 has an M ( K 1 K 2 , L * ) * -coloring. By applying Lemma 7, an M ( G , L ) -coloring of G can be achieved, leading to a contradiction.
  • d ( v ) 9 . Then we have μ ( v ) 5 . According to Lemma 20, μ * ( v ) 5 2 × 5 2 = 0 , μ * ( v ) 5 3 × 3 2 > 0 or μ * ( v ) 5 5 2 1 2 3 2 > 0 .
Consider a face f C 0 in G. By (R9), μ * ( f ) = 0 . Now, we examine μ * ( C 0 ) in G. Consider that f 3 represents the number of 3-faces that have vertices in common with C 0 and let b denote the charge transferred to C 0 from other faces by (R9). Define s as the size of [ C 0 , G C 0 ] . By Lemma 9 (i), f 3 + s 1 . If f 3 1 , then s f 3 + 1 . If f 3 = 0 , then s 1 , and so we also find s f 3 + 1 . Using rules (R8) and (R9), we can express μ * ( C 0 ) as follows:
μ * ( C 0 ) = μ ( C 0 ) + v V ( C 0 ) ( d ( v ) 4 ) f 3 + b
= d ( C 0 ) + 4 + v V ( C 0 ) ( d ( v ) 2 ) 2 d ( C 0 ) f 3 + b
= 4 d ( C 0 ) + s f 3 + b .
Since s f 3 + 1 , μ * ( C 0 ) 4 d ( C 0 ) + 1 + b 2 + b for d ( C 0 ) = 3 . If b 0 , then μ * ( C 0 ) > 0 , which completes this section.
Next, we prove that b 0 . Define μ ( f ) as the charge assigned to face f following the application of rules (R1)–(R8). Clearly, b = f F { C 0 } μ ( f ) . To begin with, let us examine the scenario where d ( f ) = 4 . In this instance, f is not impacted by the discharging procedure. Thus, μ ( f ) = μ ( f ) = d ( f ) 4 = 0 . If d ( f ) = 5 , then f is adjacent to no more than a 3-face for G contains no 7-cycle adjacent to 4- or 5-cycles. By (R1), μ ( f ) = μ ( f ) 1 2 > 0 . Now, assume f is a 6 + -face. Denote f to be adjacent to m 3-faces, meaning f has m edges shared with 3-faces. If f is a 6- or 7-face, then by (R1), μ ( f ) = μ ( f ) μ ( f ) d ( f ) × m μ ( f ) 2 d ( f ) × ( d ( f ) m ) × 2 = 0 . In the case of d ( f ) 8 , we find μ ( f ) = μ ( f ) 1 2 × m 1 4 × ( d ( f ) m ) × 2 = d ( f ) 2 4 0 by (R1). Finally, d ( f ) = 3 . If f is a 3-face, then according to Lemma 8, f could be a subgraph of an i-cluster where i [ 7 ] .
Let K be an l-cluster. Note that C 0 is not a face of K. This means | V ( K ) V ( C 0 ) | 2 . We define the initial and the middle charges of K after applying (R1)–(R8), denoted by μ ( K ) and μ ( K ) , as the sum of the initial and middle charges of l 3-faces in it. Note that the final charge of every 3-face is 0 by (R9). We check μ ( K ) 0 in the following.
  • l = 1 . If K is not internal, then μ ( K ) μ ( K ) + 1 = 0 by (R8). Thus, assume that K is internal. Assume that f i is the face adjacent to K for i = 1 , 2 , 3 with d ( f 1 ) d ( f 2 ) d ( f 3 ) . If d ( f 1 ) = 4 , then d ( f i ) 5 and d ( f i ) 6 , 7 for each i { 2 , 3 } . By (R1), each f i ( i = 2 , 3 ) gives at least 1 2 to K and μ ( K ) μ ( K ) + 1 2 × 2 0 . Thus, d ( f 1 ) 5 . By (R1), μ ( K ) μ ( K ) + 3 × 1 3 = 0 .
  • l = 2 . Assume that K is not internal. If | V ( C 0 ) V ( K ) | = 2 , then by rule (R8), μ ( K ) μ ( K ) + 1 × 2 = 0 . So, suppose | V ( C 0 ) V ( K ) | = 1 . According to Lemma 11(iii), K is adjacent to at least two 8 + -faces. Utilizing (R1) and (R8), μ ( K ) μ ( K ) + 1 + 1 2 × 2 = 0 .
    Thus, K is internal. By Lemma 11 (iii), the face adjacent to K is a 4-face or an 8 + -face and there are no more than two 4-faces. If all faces adjacent to K are 8 + -faces, then by (R1), μ ( K ) μ ( K ) + 4 × 1 2 = 0 . Thus suppose one adjacent face is a 4-face. Utilizing (R2) and Lemma 9 (iii), each 3-type vertex of K gives 1 4 to K. Then, according to (R1) and (R2), we find μ ( K ) μ ( K ) + 1 2 × 3 + 1 4 × 2 = 0 or μ ( K ) μ ( K ) + 1 2 × 2 + 1 4 × 2 + 1 4 × 2 = 0 .
  • l = 3 . If | V ( C 0 ) V ( K ) | = 2 , then K must be adjacent to at least two 8 + -faces. Utilizing (R1) and (R8), μ ( K ) μ ( K ) + 2 + 1 2 × 2 = 0 . If | V ( C 0 ) V ( K ) | = 1 , then K is adjacent to a minimum of three 8 + -faces. By applying (R1), (R3) and (R8), it can be concluded that μ ( K ) μ ( K ) + 1 + 1 2 × 3 + 1 4 × 2 0 or μ ( K ) μ ( K ) + 2 + 1 2 × 3 + 1 4 0 . Thus | V ( C 0 ) V ( K ) | = 0 . If the faces adjacent to K are all 8 + -faces, then μ ( K ) μ ( K ) + 1 2 × 5 + 1 4 × 2 = 0 by (R1) and (R3). Thus, according to Lemma 11(iii), there exists one adjacent 4-face which contains 4-type vertex of K. By applying (R1) and (R3), μ ( K ) μ ( K ) + 1 2 × 3 + 1 4 × 2 + 1 2 × 2 = 0 .
  • l = 4 . By Lemma 8, K is a graph of { H 4 1 , H 4 2 , H 4 3 , H 4 4 } shown in Figure 3. By Lemma 11 (ii), faces adjacent to K are 8 + -faces. If | V ( C 0 ) V ( K ) | = t , where t { 1 , 2 } , then C 0 gives at least t to K when K { H 4 2 , H 4 3 , H 4 4 } , t + 1 to K when K H 4 1 by (R8). If K { H 4 2 , H 4 3 , H 4 4 } , then there are at least 7 t edges of K adjacent to 8 + -faces. By (R1) and (R8), μ ( K ) μ ( K ) + ( 7 t ) × 1 2 + t 0 . If K H 4 1 , μ ( K ) μ ( K ) + ( 5 t ) × 1 2 + ( t + 1 ) 0 by (R1) and (R8).
    Now, suppose K is internal. Since H 4 1 has four 3-type 4 + -vertices and each of { H 4 2 , H 4 3 } has two 3-type 4 + -vertices, μ ( K ) μ ( K ) + 4 × 1 2 + 4 × 1 2 0 or μ ( K ) μ ( K ) + 6 × 1 2 + 2 × 1 2 0 by (R1) and (R4). Suppose K H 4 4 . If K is special, then by (R1), (R4) and Lemma 10, μ ( K ) μ ( K ) + 6 × 1 2 + 3 × 1 2 > 0 . Otherwise, μ ( K ) μ ( K ) + 6 × 1 2 + 3 2 > 0 .
  • l = 5 . By Lemma 8, K is a graph of { H 5 1 , H 5 2 } shown in Figure 3. By Lemma 11(ii), faces adjacent to K are 8 + -faces. If | V ( C 0 ) V ( K ) | = 2 , then C 0 gives 3 to K by (R8). In this case, there are four edges of K adjacent to 8 + -faces. By (R1) and (R8), μ ( K ) μ ( K ) + 3 + 4 × 1 2 0 . If | V ( C 0 ) V ( K ) | = 1 , then there are five edges of K adjacent to 8 + -faces. If K is H 5 1 , then K has a 5-type 5-vertex and μ ( K ) μ ( K ) + 2 + 5 × 1 2 + 1 > 0 . Assume K is H 5 2 . If u 1 V ( C 0 ) , then μ ( K ) μ ( K ) + 1 + 5 × 1 2 + 2 × 1 > 0 when H 5 2 is special; otherwise, μ ( K ) μ ( K ) + 1 + 5 × 1 2 + 1 2 + 3 2 > 0 . If u 1 V ( C 0 ) , then μ ( K ) μ ( K ) + 2 + 5 × 1 2 + 1 2 × 2 > 0 by (R1), (R5) and (R8).
    Thus, assume K is internal. If K H 5 1 , then K contains five 3-type bad 4 + -vertices and one 5-type 5-vertex. By (R1) and (R5), μ ( K ) μ ( K ) + 1 2 × 5 + 1 2 × 5 + 1 > 0 . Assume K is H 5 2 . If K is special, then K contains two 3-type good 5 + -vertices and one 2-type good 5 + -vertex. By (R1) and (R5), μ ( K ) μ ( K ) + 1 2 × 5 + 1 × 2 + 1 2 = 0 . Otherwise, μ ( K ) μ ( K ) + 1 2 × 5 + 3 2 + 1 2 × 2 = 0 .
  • l = 6 . By Lemma 8, K is the graph H 6 shown in Figure 3. By Lemma 11(ii), the faces adjacent to K are 8 + -faces. If | V ( C 0 ) V ( K ) | = 2 , then C 0 gives 4 to K by (R8). In this case, there are three edges of K adjacent to 8 + -faces. Due to symmetry, suppose that u 1 , u 3 V ( C 0 ) . Then u 4 is a 3-type 4 + -vertex. By (R6), u 4 gives at least 1 2 to K. By (R1) and (R8), μ ( K ) μ ( K ) + 3 × 1 2 + 4 + 1 2 = 0 . If | V ( C 0 ) V ( K ) | = 1 , then C 0 contributes at least 2 to K by (R8) and there are four edges of K adjacent to 8 + -faces. Suppose firstly, K is special. Then, K has at least two internal 5 + -vertices. Applying (R1), (R6), and (R8), we find that μ ( K ) μ ( K ) + 4 × 1 2 + 2 + 2 × 1 0 . Suppose secondly, that K is not special. Then, K contains one 4-type 5 + -vertex and one 3-type bad 4 + -vertex. Utilizing (R1), (R6), and (R8), μ ( K ) μ ( K ) + 4 × 1 2 + 2 + 3 2 + 1 2 0 .
    Thus, assume K is internal. If K is special, then K contains two 3-type good 5 + -vertices and one 4-type good 5 + -vertex by Lemma 10. If u 1 , u 4 A 1 , then d ( u 5 ) 6 by Lemma 18. This means u 5 is a 4-type good 6 + -vertex and K contains two vertices of A 1 . In this case, μ ( K ) μ ( K ) + 4 × 1 2 + 2 + 2 × 1 = 0 by (R1) and (R6). Then, assume u 1 A 1 . By (R6), u 1 gives at least 3 2 to K. By (R1) and (R6), μ ( K ) μ ( K ) + 4 × 1 2 + 1 + 2 × 3 2 = 0 . Suppose K is not special. In this situation, u i is classified as a 4-type 5 + -vertex for each i { 3 , 5 } . By (R1) and (R6), μ ( K ) μ ( K ) + 4 × 1 2 + 2 × 3 2 + 2 × 1 2 = 0 .
  • l = 7 . By Lemma 8, K is the graph H 7 shown in Figure 3. By Lemma 11(ii), faces adjacent to K are 8 + -faces. If | V ( C 0 ) V ( K ) | = 2 , then C 0 gives 5 to K by (R8) and according to Lemma 10, K contains one 4-type 5 + -vertex. If | V ( C 0 ) V ( K ) | = 1 , then C 0 contributes 3 to K and K contains two 4-type 5 + -vertices similarly. By (R1), (R7) and (R8), μ ( K ) μ ( K ) + 5 + 2 × 1 2 + 3 2 > 0 or μ ( K ) μ ( K ) + 3 + 3 × 1 2 + 2 × 3 2 > 0 . Thus assume K is internal. By Lemma 10, d ( u i ) 5 for each i { 1 , 4 , 5 } . By Lemma 19(ii), there are at most two vertices of { u 1 , u 4 , u 5 } in A 2 { the special 8-vertex and K is type II}. If any vertex of { u 1 , u 4 , u 5 } is not in A 2 { the special 8-vertex and K is type II}, then μ ( K ) μ ( K ) + 3 × 1 2 + 3 × 2 > 0 . If there is only one vertex in A 2 { the special 8-vertex and K is type II}, then μ ( K ) μ ( K ) + 3 × 1 2 + 3 2 + 2 × 2 = 0 . Thus assume there are two vertices in A 2 { the special 8-vertex and K is type II}, say u 4 , u 5 . By Lemma 19(i), d ( u 1 ) 7 . By (R1) and (R7), μ ( K ) μ ( K ) + 3 × 1 2 + 2 × 3 2 + 5 2 = 0 .

5. Conclusions

DP-k-coloring has gained significant attention as a generalization of k-choosability. We demonstrate that any planar graph G, which does not have a Near-bow-tie as an induced subgraph and lacks 7-cycles adjacent to 4-cycles or 5-cycles, is DP-4-colorable. If the planar graph G does not have 7-cycles and a butterfly, then it still satisfies our condition. Thus, our results generalize the findings of Kim et al. [20] and promote research in the area of DP-4-colorability for planar graphs. Future research can focus on determining the DP-4-colorability on planar graphs such as χ D P ( G ) = 4 for any planar graph G without 7-cycles adjacent to 4-cycles.

Author Contributions

F.Y., X.L. and Z.H. were involved in the conception and design of the study, and they all reviewed and approved the final version of the manuscript. The authors attest that all data supporting the findings of this research are included within the article. All authors have read and agreed to the published version of the manuscript.

Funding

This research received funding from the NSF of China under grant number 12031018 and the NSF of China under grant number 12261094.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Acknowledgments

The author would like to thank the anonymous reviewers and the editor for their careful reviews and constructive suggestions to help us improve the quality of this paper.

Conflicts of Interest

The authors declare no conflicts of interest.

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Figure 1. H 1 and H 2 are two distinct covers of ( C 4 , L ) with | L ( v i ) | = 2 for each i [ 4 ] . (a) 4-cycle C 4 . (b) H 1 . (c) H 2 .
Figure 1. H 1 and H 2 are two distinct covers of ( C 4 , L ) with | L ( v i ) | = 2 for each i [ 4 ] . (a) 4-cycle C 4 . (b) H 1 . (c) H 2 .
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Figure 2. Near-bow-tie and butterfly. (a) Near-bow-tie, where v is a 7-vertex. (b) butterfly.
Figure 2. Near-bow-tie and butterfly. (a) Near-bow-tie, where v is a 7-vertex. (b) butterfly.
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Figure 3. All non-isomorphic k-clusters for each k { 4 , 5 , 6 , 7 } .
Figure 3. All non-isomorphic k-clusters for each k { 4 , 5 , 6 , 7 } .
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Figure 4. Different cover M ( K 3 , L ) of ( K 3 , L ) , where L v i is a clique for each i [ 3 ] . (a) E M is a union of two 3-cycles. (b) E M is not a union of two 3-cycles.
Figure 4. Different cover M ( K 3 , L ) of ( K 3 , L ) , where L v i is a clique for each i [ 3 ] . (a) E M is a union of two 3-cycles. (b) E M is not a union of two 3-cycles.
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Figure 5. Special or poor vertex: all K 1 is special and internal and K 2 is special. (a) Special 6-vertex v. (b) Poor 6-vertex v. (c) Poor 7-vertex v when u , w A 1 . (d) Special 7-vertex v when u A 1 and w A 2 .
Figure 5. Special or poor vertex: all K 1 is special and internal and K 2 is special. (a) Special 6-vertex v. (b) Poor 6-vertex v. (c) Poor 7-vertex v when u , w A 1 . (d) Special 7-vertex v when u A 1 and w A 2 .
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Yang, F.; Li, X.; Huang, Z. DP-4-Colorability on Planar Graphs Excluding 7-Cycles Adjacent to 4- or 5-Cycles. Mathematics 2025, 13, 190. https://doi.org/10.3390/math13020190

AMA Style

Yang F, Li X, Huang Z. DP-4-Colorability on Planar Graphs Excluding 7-Cycles Adjacent to 4- or 5-Cycles. Mathematics. 2025; 13(2):190. https://doi.org/10.3390/math13020190

Chicago/Turabian Style

Yang, Fan, Xiangwen Li, and Ziwen Huang. 2025. "DP-4-Colorability on Planar Graphs Excluding 7-Cycles Adjacent to 4- or 5-Cycles" Mathematics 13, no. 2: 190. https://doi.org/10.3390/math13020190

APA Style

Yang, F., Li, X., & Huang, Z. (2025). DP-4-Colorability on Planar Graphs Excluding 7-Cycles Adjacent to 4- or 5-Cycles. Mathematics, 13(2), 190. https://doi.org/10.3390/math13020190

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