H -Irregularity Strengths of Plane Graphs

: Graph labeling is the mapping of elements of a graph (which can be vertices, edges, faces or a combination) to a set of numbers. The mapping usually produces partial sums (weights) of the labeled elements of the graph, and they often have an asymmetrical distribution. In this paper, we study vertex–face and edge–face labelings of two-connected plane graphs. We introduce two new graph characteristics, namely the vertex–face H -irregularity strength and edge–face H -irregularity strength of plane graphs. Estimations of these characteristics are obtained, and exact values for two families of graphs are determined.


Introduction
All graphs G = (V(G), E(G)) considered in this paper are simple graphs with a vertex set V(G) and an edge set E(G). If we consider a plane graph G = (V(G), E(G), F(G)), then F(G) is its face set. For the notation and terminology not defined here, see the work presented in [1].
Graph labeling is the mapping of elements of a graph (which can be vertices, edges, faces or a combination) to a set of numbers. The mapping usually produces partial sums (weights) of the labeled elements of the graph, and they often have an asymmetrical distribution.
For a given edge k-labeling α : E(G) → {1, 2, . . . , k}, where k is a positive integer, the associated weight of a vertex x ∈ V(G) is w α (x) = ∑ xy∈E(G) α(xy). Such a labeling α is called irregular if w α (x) = w α (y) for every pair x, y of vertices of G. The smallest integer k for which an irregular labeling of G exists is known as the irregularity strength of G and is denoted by s(G). The notion of irregularity strength was introduced by Chartrand et al. in [2]. It is known that s(G) ≤ |V(G)| − 1 for graphs with no component with an order of at most two (see [3,4]). This upper bound was gradually improved by Cuckler and Lazebnik in [5], Przybyło in [6], Kalkowski, Karonski and Pfender in [7] and recently by Majerski and Przybylo in [8]. Other interesting results regarding the irregularity strength can be found in [9][10][11][12].
Let H 1 , H 2 , . . . , H t be all subgraphs in G isomorphic to a given graph H. If every edge of E(G) belongs to at least one of the subgraphs H i , i = 1, 2, . . . , t, we say that graph G admits an H-covering.
Motivated by the irregularity strength of a graph G, Ashraf et al. in [13] introduced two parameters: edge H-irregularity strength ehs(G, H) and vertex H-irregularity strength vhs(G, H), as a natural extension of the parameter s(G). The bounds of the parameters ehs(G, H) and vhs(G, H) are estimated in [13], and the exact values of the edge (vertex) H-irregularity strength are determined for several families of graphs, namely paths, ladders and fan graphs.
If we consider a plane graph, it is natural to label not only the vertices and edges of the plane graph but also its faces. Motivated by the edge (vertex) H-irregularity strength and the entire face irregularity strength of plane graphs introduced in [14], we study a vertex-face (edge-face) H-irregularity strength of two-connected plane graphs, which is a natural extension of the edge (vertex) H-irregularity strength of graphs.
A plane graph is a particular drawing of a planar graph on the Euclidean plane. Suppose that G = (V(G), E(G), F(G)) is a two-connected plane graph with a vertex set V(G), an edge set E(G) and a face set F(G), where F int (G) denotes the set of internal faces of G. In this paper, we consider only such subgraphs H of a plane graph G that every inner face of H is also an inner face of G.
For a subgraph H ⊆ G under vertex-face k-labeling ϕ : Vertex-face k-labeling ϕ is said to represent the H-irregular vertex-face k-labeling of the plane graph G admitting an H-covering if for every two different subgraphs H and H isomorphic to H there is w ϕ (H ) = w ϕ (H ). The vertex-face H-irregularity strength of a plane graph G, denoted by vfhs(G, H), is the smallest integer k such that G has H-irregular vertex-face k-labeling.
Similarly, edge-face k-labeling ψ is said to be an H-irregular edge-face k-labeling of the plane graph G admitting an H-covering if for every two different subgraphs H and H isomorphic to H there is w ψ (H ) = w ψ (H ). The edge-face H-irregularity strength of a plane graph G, denoted by efhs(G, H), is the smallest integer k such that G has H-irregular edge-face k-labeling.
The main aim of this paper is to estimate the lower bound and an upper bound for the parameters vfhs(G, H) and efhs(G, H). We determine the exact values of the vertex-face (edge-face) H-irregularity strengths for some graphs in order to prove the sharpness of the lower bounds of these graph invariants.

Lower Bounds
The next theorem gives a lower bound for the vertex-face H-irregularity strength for plane graphs.
This implies that By applying a similar reasoning, we get a lower bound for the edge-face H-irregularity strength of plane graphs as follows.
Theorem 2. Given a two-connected plane graph G = (V(G), E(G), F(G)) admitting an Hcovering with t subgraphs isomorphic to H, it holds that The lower bounds in Theorems 1 and 2 are tight. This can be seen from the following two theorems, which determine the exact values of the vertex-face and edge-face ladderirregularity strengths for ladders. First, we recall the definition and properties of a ladder.
Let L n ∼ = P n P 2 , n ≥ 2 be a ladder with the vertex set V( Thus, the ladder L n contains 2n vertices, 3n − 2 edges, and a number of four-sided face of n − 1, and the ladder has one 2n-sided face. Proof. For every m, 2 ≤ m ≤ n, the ladder L n admits an L m -covering with exactly n − m + 1 subgraphs L j m , j = 1, 2, . . . , It is clearly visible that every edge of L n belongs to at least one ladder L j m if m = 2, 3, . . . , n.
Using Theorem 1, we find that vfhs(L n , L m ) ≥ (2m To prove the equality, it suffices to show the existence of an optimal L m -irregular vertex-face k-labeling of ladder L n . For every m = 2, 3, . . . , n we define a vertex-face k-labeling ϕ m : V(L n ) ∪ F int (L n ) → {1, 2, . . . , k} in the following way: We can see that under the vertex-face labeling ϕ m , all vertex and face labels are at most k. For the L m -weight of the ladder L j m , j = 1, 2, . . . , n − m + 1, under the vertex-face labeling ϕ m , m = 2, 3, . . . , n, we get Consider the difference of weights of subgraphs L j+1 m and L j m , for j = 1, 2, . . . , n − m, as follows: Since all vertex labels and face labels under the vertex-face labeling ϕ m form nondecreasing sequences and w ϕ m (L j+1 m ) = w ϕ m (L j m ) + 1 for every m = 2, 3, . . . , n, j = 1, 2, . . . , n − m, it follows that the labeling ϕ m is an optimal L m -irregular vertex-face klabeling of L n . Thus, we arrive at the desired result.  Proof. Let the L m -covering of the ladder L n be defined by subgraphs L j m , j = 1, 2, . . . , n − m + 1 and m = 2, 3, . . . , n. From Theorem 2, we find that efhs(L n , L m ) ≥ (3m . . , n, as follows: Evidently, the labeling ψ m is assigned to the edges and faces of L n the labels less than or equal to k. If the L m -weight of the ladder L j m under the edge-face labeling ψ m is given by the form then, for the difference of weights of subgraphs L j+1 m and L j m , for j = 1, 2, . . . , n − m, m = 2, 3, . . . , n, we get This proves that w ψ m (L    H, and S i is their maximum common subgraph. Let ϕ be an optimal H-irregular vertex-face labeling of G. Then for j = 1, 2, . . . , r i the H S i j -weights given in the form are all distinct, and each of them contains the value The largest among these H S i j -weights must be at least does not have any impact on the estimation of the vertex-face irregularity strength. The term |V(H\S i )| + |F int (H\S i )| + r i − 1, for i = 1, 2, . . . , s is the sum of |V(H\S i )| + |F int (H\S i )| labels, and therefore, by Theorem 1, at least one label is lower-bounded by 1 + (r i − 1)/(|V(H\S i )| + |F int (H\S i )|) . Consequently, we deduce the desired inequality.
Similarly, we get a lower bound for the edge-face H-irregularity strength for plane graphs as follows. The sharpness of lower bounds from Theorems 5 and 6 results from the following two theorems, which determine the exact values of the vertex-face and edge-face H-irregularity strengths for some graphs.
Let G i , i = 1, 2 be a connected graph. Fix a vertex in G i , say v i . If we identify the vertices v 1 and v 2 , the resulting graph can be denoted by the symbol A(G 1 (v 1 ), G 2 (v 2 )). Let F n be a fan graph with the center w and let G be an arbitrary two-connected plane graph with a fixed vertex v belonging to the boundary of its external face. Now, we insert the fan graph F n into the external face of G and identify the vertices w and v. The resulting graph A(F n (w), G(v)) = A(F n , G) is a two-connected plane graph with n + |V(G)| vertices, 2n + |E(G)| − 1 edges, n + |F int (G)| − 1 internal faces and one external face. Observe that the graph operation has no impact on the number and also size of internal faces in graph G as the subgraph of A(F n , G).  Proof. If the graph G is not isomorphic to F k for every k > m, then the graph A(F n , G), n ≥ 2 admits an A(F m , G)-covering with exactly n − m + 1 graphs A(F m , G), 2 ≤ m ≤ n. Since every graph A(F m , G) j contains the graph G ∼ = S 1 with the vertex set V(S 1 ) = V(G), edge set E(S 1 ) = E(G) and face set F int (S 1 ) = F int (G) as the maximum common subgraph, it follows that |V(A(F m , G)\S 1 )| = m, |F int (A(F m , G)\S 1 )| = m − 1, r 1 = n − m + 1, and from Theorem 5, we have vfhs (A(F n , G), A(F m , G)) ≥ 1 + To show that k = (m + n − 1)/(2m − 1) is an upper bound for the vertex-face A(F m , G)-irregularity strength of A(F n , G), it suffices to prove the existence of an optimal vertex-face k-labeling ϕ m : V(A(F n , G)) ∪ F int (A(F n , G)) → {1, 2, . . . , k}. For m = 2, 3, . . . , n, we define the function ϕ m in the following way: Indeed, it is readily seen that all vertex and face labels are at most k. Since the A(F m , G)weight of the graph A(F m , G) j , under the vertex-face labeling ϕ m , is given by the form then, for the difference of weights of subgraphs A(F m , G) j+1 and A(F m , G) j , for j = 1, 2, . . . , n − m, m = 2, 3, . . . , n, we have for e ∈ E(G), Observe that all vertex and face labels under the labeling ψ m are at most k. Denote the A(F m , G)-weight of the graph A(F m , G) j , under the edge-face labeling ψ m , by For the difference of weights of subgraphs A(F m , G) j+1 and A(F m , G) j , for j = 1, 2, . . . , n − m, m = 2, 3, . . . , n, we get We can see that w ψ m (A(F m , G) j ) < w ψ m (A(F m , G) j+1 ) for every j = 1, 2, . . . , n − m and m = 2, 3, . . . , n. Thus, the labeling ψ m is a desired A(F m , G)-irregular edge-face k-labeling of A(F n , G). Figure 4 depicts an A(F 3 , G)-irregular edge-face two-labeling of A(F 9 , G), where G is a two-connected plane graph, G ∼ = F k for every k > 3, and w is the common vertex of the fan F 9 and G. The weights of the subgraphs A(F 3 , G) j , j = 1, 2, . . . , 7 constitute the set of consecutive integers {7 + |E(G)| + |F int (G)|, 8

Upper Bounds
The next theorem gives upper bounds of the parameters vfhs(G, H) and efhs(G, H) and shows that these graph invariants are always finite. Proof. Consider a plane graph G admitting the H-covering given by subgraphs H 1 , H 2 , . . . , H t . Denote the internal faces of G arbitrarily by the symbols f 1 , f 2 , . . . , f |F int (G)| .
First, we define a vertex-face 2 |F int (G)|−1 -labeling ϕ of G in the following way.
The associated H-weights are the sums of all vertex labels and face labels of vertices and faces in the given subgraph. Thus, for j = 1, 2, . . . , t we have As we have |V(H j )| = |V(H)| for every j = 1, 2, . . . , t, to prove that the H-weights are all distinct, it is enough to show that the sums ∑ |F int (G)| i=1 θ i,j 2 i−1 are distinct for every j = 1, 2, . . . , t. However, this is evident if we consider that the ordered (|F int (G)|)-tuple (θ |F int (G)|,j θ |F int (G)|−1,j . . . θ 2,j θ 1,j ) corresponds to binary code representation of the sum (1).