On Fall-Colorable Graphs

: A fall k -coloring of a graph G is a proper k -coloring of G such that each vertex has at least one neighbor in each of the other color classes. A graph G which has a fall k -coloring is equivalent to having a partition of the vertex set V ( G ) in k independent dominating sets. In this paper, we first prove that for any fall k -colorable graph G with order n , the number of edges of G is at least ( n ( k − 1 ) + r ( k − r )) /2, where r ≡ n ( mod k ) and 0 ≤ r ≤ k − 1, and the bound is tight. Then, we obtain that if G is k -colorable ( k ≥ 2) and the minimum degree of G is at least k − 2 k − 1 n , then G is fall k -colorable and this condition of minimum degree is the best possible. Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k -colorable, where k ≥ 3. Finally, we show that there exist an infinite family of fall k -colorable planar graphs for k ∈ { 5,6 } .


Introduction
In this paper, we only consider simple and undirected graphs.For a graph G = (V(G), E(G)), we use V(G) and E(G) to represent the sets of vertices and edges of G, respectively.We use d G (v) to represent the degree of a vertex v ∈ V(G), that is, the number of neighbors of v in G.If d G (v) = r for any v ∈ V(G), then the graph G is called an r-regular graph.For a vertex v ∈ V(G), let N G (v) = {u : uv ∈ E(G)} and N G [v] = N G (v) ∪ {v} denote the open neighborhood and the closed neighborhood of v, respectively.The maximum degree and minimum degree of G are denoted by ∆(G) and δ(G), respectively.When no confusion can arise, N G (v), N G [v], ∆(G), and δ(G) are simplified by N(v), N[v], ∆, and δ, respectively.A plane graph is a graph drawn in the plane such that its edges intersect only at their ends; a planar graph is a graph that can be drawn as a plane graph.
Let G be a graph.A (proper) k-coloring f of G is a mapping from V(G) to {1, 2, . . ., k} such that f (u) ̸ = f (v) for any uv ∈ E(G).Hence, a k-coloring can be regarded as a partition {V 1 , V 2 , . . ., V k } of V(G), where V i denotes the set of vertices assigned color i, and is called a color class of f , where i = 1, 2, . . ., k.If a graph G admits a k-coloring, the G is called k-colorable.The minimum number k such that G is k-colorable is called the chromatic number of G and is denoted by χ(G). Let then the vertex v is called colorful with respect to f .Furthermore, the coloring f is called colorful whenever each of its color classes contains at least one colorful vertex.The maximum order of a colorful coloring of a graph G is called the b-chromatic number of G, and is denoted by φ(G).A fall k-coloring of a graph G is a k-coloring of G such that every vertex is colorful.
The problem of b-chromatic numbers was introduced by Irving and Manlove in 1999 [1] and studied extensively in the literature (see the survey in [2]), whereas fall coloring was introduced in [3] and studied in [4][5][6].It follows from [6] that fall coloring strongly chordal graphs is doable in polynomial time, even with an unbounded number of colors.
A dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is either in S or has at least one neighbor in S. If S is a dominating set and independent, then S is an independent dominating set (IDS) of G.The independent domination number γ i (G) is the minimum cardinality of an IDS of G.A graph G has a fall k-coloring if and only if V(G) can be partitioned into k independent dominating sets [7].
Note that a graph may have no fall coloring.For instance, the cycle C n has a fall coloring only when n is a multiple of three or is even [3].Hence, determining which graphs are fall-colorable is an interesting problem.In fact, in 1976 Cockayne and Hedetniemi [7] first studied fall-colorable graphs but used another term, indominable graphs.They found several families of graphs which have fall colorings.
In this paper, we further discuss fall-colorable graphs.First, the size of a k-colorable graph is determined, including the boundaries.Then, a sufficient condition of a graph to be k-colorable (k ≥ 2) is proposed and the tightness of this condition is discussed.Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall kcolorable, where k ≥ 3. Finally, we show that there exist an infinite family of fall k-colorable planar graphs for k ∈ {5, 6} and find some sufficient conditions for a maximal planar graph to be fall-colorable.
For other notations and terminologies in graph theory, we refer to [8].

Some Properties of Fall-Colorable Graphs
In this section, we discuss some properties of fall k-colorable graphs.The following, Lemmas 1 and 2, can be obtained straight from previous studies, such as [3,7].

Lemma 1 ([3]
).Let G be a fall k-colorable graph and f a fall k-coloring.We have the following: (ii) The subgraph induced by the union of any r color classes under f is fall r-colorable, where r ≤ k.

Lemma 2 ([7]). A graph G is fall k-colorable if and only if G has a k-coloring such that the subgraph induced by the union of any two color classes has no isolated vertices.
Theorem 1.Let G be a fall k-colorable graph of order n.Then, where r ≡ n (mod k) and 0 ≤ r ≤ k − 1.
Without a loss of generality, we assume that For any two color classes V i and V j with i < j, by Theorem 2, we know that the subgraph G i,j induced by Now, we prove that if However, this contradicts the minimality of and n ′ i = n i for any i ∈ {1, 2, . . ., k} \ {b, c + 1}.Similar to the former case, we can obtain Let n = kt + r, where r ≡ n (mod k) and 0 ≤ r ≤ k − 1.Now, we consider the case of ∑ k i=1 i • n i as the minimum.Note that ( Together with Formulae (1) and ( 2), we have Theorem 2. For any fall k-colorable graph G with order n, if G is (k − 1)-regular, then n ≡ 0 (mod k).Moveover, for any fall k-coloring f of G, each color class of f has exactly n k vertices.
Proof.Let V i be any color class of the fall k-coloring f of G.Then, for any two vertices u and v in V i ; we can obtain -regular, we can deduce that x is adjacent to at most k − 2 color classes, which implies that x is not a colorful vertex of f ; this is a contradiction.
. ., t, we have n = kt and so n ≡ 0 (mod k).Note that |V i | = t = n k , we can discover that each color class of f has exactly n k vertices.

A Sufficient Condition
In 2010, Balakrishnan and Kavaskar [9] showed that any graph G with δ(G) ≥ |V(G)| − 2 admits a fall coloring.In this section, we improve this result by relaxing the con- | for any k ≥ 2 and prove that the condition of δ(G) is the best possible.First, we give a useful lemma obtained by Zarankiewicz [10]: Lemma 3 ([10]).Let G be a k-colorable graph with n vertices and δ(G) > k−2 k−1 n, where k ≥ 2. We have χ(G) = k.

Theorem 3. Let G be a k-colorable graph with n vertices and δ(
k−1 n = 0 and G has no isolated vertices.Hence, G is fall 2-colorable.Now, assume that k ≥ 3. Let v be an arbitrary vertex of G and G v be the subgraph of Note that G is k-colorable, so G v is (k − 1)-colorable.Hence, by Lemma 3, we can see that χ(G v ) = k − 1, which yields that | f (N G (v))| = k − 1 for any k-coloring f of G.That is to say, v is a colorful vertex with respect to f .Since v is an arbitrary vertex of G, we can deduce that f is a fall k-coloring of G. Hence, the graph G is fall k-colorable.Now, we show that the condition δ(G) > k−2 k−1 n in Theorem 3 is the best possible.We will construct a family of graphs that are not fall k-colorable, G ℓ , with δ(G ℓ ) = k−2 k−1 |V(G ℓ )|.We use K n to denote the complete graph of order n and use T r,s to denote the complete r-partite graph with s vertices in each class, where r ≥ 2. The join of two graphs G and H, denoted as G ∨ H, is the graph obtained from the disjointed union of G and H, and we add edges joining every vertex of G to every vertex of H.
For any k ≥ 3 and ℓ ≥ 1, let , and For example, when k = 4 and ℓ = 1, the graph G ℓ is shown in Figure 1.
) is not a colorful vertex with respect to f .So, G ℓ is not fall k-colorable.

Complexity
The problem of determining whether a graph is fall k-colorable (k ≥ 3) has been shown to be NP-complete [3,[11][12][13].In this section, we give a simple proof for the NP-complete result of the FALL k-COLORABLE problem, which is defined as follows: FALL k-COLORABLE: Instance: Given a graph G = (V, E) and a positive integer k.Question: Is G fall k-colorable?k-COLORABLE: Instance: Given a graph G = (V, E) and a positive integer k.Question: Is G k-colorable?
It is well known that the k-COLORABLE problem is NP-hard for any k ≥ 3 [14].We will prove that the fall k-colorable problem is NP-hard by using a reduction from the k-COLORABLE problem.
Theorem 4. FALL k-COLORABLE is NP-complete for any k ≥ 3.
Proof.We show that the FALL k-COLORABLE problem is NP-complete by a reduction from k-COLORABLE.For any graph G of order n with the vertex set {v 1 , v 2 , • • • , v n }, we construct a graph G ′ as follows: First, take n copies Then, add these n copies of K k to G and identify v i and a vertex of K i k into a single vertex, where Furthermore, the k-COLORABLE problem remains NP-hard under several restrictions.Garey and Johnson [15] proved the following: Lemma 4 ( [15]).Three-COLORABLE is NP-complete even when restricted to planar graphs with a maximum degree of four.
By Lemma 4 and using a similar approach to that in the proof of Theorem 4, we can obtain the following result.
Corollary 1. FALL 3-COLORABLE is NP-complete even when restricted to planar graphs with a maximum degree of six.

Fall Colorings of Planar Graphs
In this section, we discuss the fall colorings of planar graphs.Since δ(G) ≤ 5 for any planar graph G, it follows from Lemma 1 (i) that ψ f (G) ≤ 6.In [7], Cockayne and Hedetniemi found that each uniquely k-colorable graph is fall k-colorable.Note that for any integer k ≤ 4, there exist an infinite family of planar graphs that are uniquely kcolorable [16][17][18][19][20][21], but uniquely five-colorable planar graphs do not exist [18].Hence, there exist an infinite family of planar fall k-colorable graphs for any k ≤ 4. Now, we show that there also exist an infinite family of planar fall k-colorable graphs for k ∈ {5, 6}.
We can see that the icosahedron G 12 in Figure 2, which is a planar graph, has a fall five-coloring (Figure 2a) and a fall six-coloring (Figure 2b).Theorem 5.There exist an infinite family of planar fall k-colorable graphs for k ∈ {5, 6}.
Proof.From the icosahedron G 12 , we can construct a family of graphs {H i } as follows: (1) H 1 = G 12 ; (2) For integer i ≥ 2, H i can be obtained by embedding a copy of G 12 in some interior face of H i−1 and identifying the boundaries of this face and the exterior face of G 12 .
It can be checked that H i is a planar graph of order 9i + 3. Note that every threecoloring of the exterior triangle of G 12 can be extended to a fall five-coloring by Figure 2a or a fall six-coloring by Figure 2b of G 12 .We can recursively obtain a fall five-coloring and a fall six-coloring of H i .Hence, for any integer i, H i is a planar fall k-colorable graph for k ∈ {5, 6}.Now, we discuss the fall colorings of maximal planar graphs.A planar graph G is maximal if G + uv is not planar for any two nonadjacent vertices u and v of G.For example, the icosahedron G 12 in Figure 2 is a maximal planar graph.Theorem 6 ([12]).If a maximal planar graph G is three-colorable, then G is fall three-colorable.
Since a maximal planar graph G is three-colorable if and only if every vertex in G has an even degree [22,23], we can obtain the following result: Corollary 2. Let G be a maximal planar graph.If each vertex in G has an even degree, then G is fall three-colorable.Theorem 7. Let G be a maximal planar graph.If each vertex in G has an odd degree, then G is fall four-colorable.
Proof.It follows from the Four Color Theorem [24,25] that G is four-colorable.Let f be a four-coloring of G. Since G is a maximal planar graph, we know that the neighbors of each vertex v form a cycle C v of order d G (v).Note that v has an odd degree in G. Hence, C v contains three colors under the coloring f , that is, v is colorful with respect to f .So, f is a fall four-coloring of G.

Conclusions and Open Problems
In this paper, we first show that |E(G)| ≥ (n(k − 1) + r(k − r))/2 for any fall kcolorable graph G with order n, where r ≡ n (mod k) and 0 ≤ r ≤ k − 1, and this bound is tight.Then, we obtain that if G is k-colorable (k ≥ 2) and the minimum degree δ(G) ≥ k−2 k−1 n, then G is fall k-colorable and this condition of the minimum degree is the best possible.
Moreover, we give a simple proof for an NP-hard result of determining whether a graph is fall k-colorable, where k ≥ 3.
For any outerplane graph G, note that the minimum degree δ(G) ≤ 2. If G has a fall k-coloring, then by Lemma 1 we have k ≤ δ(G) + 1 ≤ 3.