Clique coloring of dense random graphs

The clique chromatic number of a graph is the minimum number of colors in a vertex coloring so that no maximal (with respect to containment) clique is monochromatic. We prove that the clique chromatic number of the binomial random graph is, with high probability, . This settles a problem of McDiarmid, Mitsche, and Prałat who proved that it is with high probability.     

Improper coloring of graphs on surfaces

A graph is -colorable if its vertex set can be partitioned into sets , such that for each , the subgraph of induced by has maximum degree at most . The Four Color Theorem states that every planar graph is -colorable, and a classical result of Cowen, Cowen, and Woodall shows that every planar graph is -colorable. In this paper, we extend both of these results to graphs on surfaces. Namely, we show that every graph embeddable on a surface of Euler genus is -colorable and -colorable. Moreover, these graphs are also -colorable and -colorable. We also prove that every triangle-free graph that is embeddable on a surface of Euler genus is -colorable. This is an extension of Grötzsch's Theorem, which states that triangle-free planar graphs are -colorable. Finally, we prove that every graph of girth at least 7 that is embeddable on a surface of Euler genus is -colorable. All these results are best possible in several ways as the girth condition is sharp, the constant maximum degrees cannot be improved, and the bounds on the maximum degrees depending on are tight up to a constant multiplicative factor.     

Optimal acyclic edge‐coloring of cubic graphs

An acyclic edge‐coloring of a graph is a proper edge‐coloring such that the subgraph induced by the edges of any two colors is acyclic. The acyclic chromatic index of a graph G is the smallest number of colors in an acyclic edge‐coloring of G. We prove that the acyclic chromatic index of a connected cubic graph G is 4, unless G is K₄ or K_3,3; the acyclic chromatic index of K₄ and K_3,3 is 5. This result has previously been published by Fiam?ík, but his published proof was erroneous.     

Acyclic list 7‐coloring of planar graphs

The acyclic list chromatic number of every planar graph is proved to be at most 7. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 83–90, 2002     

Acyclic edge coloring of 2‐degenerate graphs

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). A graph is called 2‐degenerate if any of its induced subgraph has a vertex of degree at most 2. The class of 2‐degenerate graphs properly contains series–parallel graphs, outerplanar graphs, non ? regular subcubic graphs, planar graphs of girth at least 6 and circle graphs of girth at least 5 as subclasses. It was conjectured by Alon, Sudakov and Zaks (and much earlier by Fiamcik) that a′(G)?Δ + 2, where Δ = Δ(G) denotes the maximum degree of the graph. We prove the conjecture for 2‐degenerate graphs. In fact we prove a stronger bound: we prove that if G is a 2‐degenerate graph with maximum degree Δ, then a′(G)?Δ + 1. © 2010 Wiley Periodicals, Inc. J Graph Theory 69: 1–27, 2012     

Acyclic edge coloring of graphs with maximum degree 4

An acyclic edge coloring of a graph is a proper edge coloring such that there are no bichromatic cycles. The acyclic chromatic index of a graph is the minimum number k such that there is an acyclic edge coloring using k colors and is denoted by a′(G). It was conjectured by Alon, Sudakov, and Zaks that for any simple and finite graph G, a′(G)?Δ + 2, where Δ=Δ(G) denotes the maximum degree of G. We prove the conjecture for connected graphs with Δ(G)?4, with the additional restriction that m?2n?1, where n is the number of vertices and m is the number of edges in G. Note that for any graph G, m?2n, when Δ(G)?4. It follows that for any graph G if Δ(G)?4, then a′(G)?7. © 2009 Wiley Periodicals, Inc. J Graph Theory 61: 192–209, 2009     

Dynamic list coloring of bipartite graphs

A dynamic coloring of a graph is a proper coloring of its vertices such that every vertex of degree more than one has at least two neighbors with distinct colors. The least number of colors in a dynamic coloring of G, denoted by χ₂(G), is called the dynamic chromatic number of G. The least integer k, such that if every vertex of G is assigned a list of k colors, then G has a proper (resp. dynamic) coloring in which every vertex receives a color from its own list, is called the choice number of G, denoted by ch(G) (resp. the dynamic choice number, denoted by ch₂(G)). It was recently conjectured (Akbari et al. (2009) [1]) that for any graph G, ch₂(G)=max(ch(G),χ₂(G)). In this short note we disprove this conjecture. We first give an example of a small planar bipartite graph G with ch(G)=χ₂(G)=3 and ch₂(G)=4. Then, for any integer k≥5, we construct a bipartite graph G_k such that ch(G_k)=χ₂(G_k)=3 and ch₂(G)≥k.     

Star coloring bipartite planar graphs

A star coloring of a graph is a proper vertex‐coloring such that no path on four vertices is 2‐colored. We prove that the vertices of every bipartite planar graph can be star colored from lists of size 14, and we give an example of a bipartite planar graph that requires at least eight colors to star color. © 2008 Wiley Periodicals, Inc. J Graph Theory 60: 1–10, 2009     

A conjecture of Borodin and a coloring of Grünbaum

In 1976, Borodin conjectured that every planar graph has a 5‐coloring such that the union of every k color classes with 1 ≤ k ≤ 4 induces a (k—1)‐degenerate graph. We prove the existence of such a coloring using 18 colors. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:139–147, 2008     

Star coloring of sparse graphs

A proper coloring of the vertices of a graph is called a star coloring if the union of every two color classes induces a star forest. The star chromatic number χ_s(G) is the smallest number of colors required to obtain a star coloring of G. In this paper, we study the relationship between the star chromatic number χ_s(G) and the maximum average degree Mad(G) of a graph G. We prove that:

• 1. If G is a graph with , then χ_s(G)≤4.
• 2. If G is a graph with and girth at least 6, then χ_s(G)≤5.
• 3. If G is a graph with and girth at least 6, then χ_s(G)≤6.

These results are obtained by proving that such graphs admit a particular decomposition into a forest and some independent sets. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 201–219, 2009     

The generalized acyclic edge chromatic number of random regular graphs

The r‐acyclic edge chromatic number of a graph is defined to be the minimum number of colors required to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min(|C|, r) colors. We show that (r ? 2)d is asymptotically almost surely (a.a.s.) an upper bound on the r‐acyclic edge chromatic number of a random d‐regular graph, for all constants r ≥ 4 and d ≥ 2. © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 101–125, 2006     

Some results on the incidence coloring number of a graph

This paper proves that if G is a cubic graph which has a Hamiltonian path or G is a bridgeless cubic graph of large girth, then its incidence coloring number is at most 5. By relating the incidence coloring number of a graph G to the chromatic number of G², we present simple proofs of some known results, and characterize regular graphs G whose incidence coloring number equals Δ(G)+1.     

The acyclic edge chromatic number of a random d‐regular graph is d + 1

We prove the theorem from the title: the acyclic edge chromatic number of a random d‐regular graph is asymptotically almost surely equal to d + 1. This improves a result of Alon, Sudakov, and Zaks and presents further support for a conjecture that Δ(G) + 2 is the bound for the acyclic edge chromatic number of any graph G. It also represents an analog of a result of Robinson and the second author on edge chromatic number. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 69–74, 2005     

On the adjacent vertex-distinguishing acyclic edge coloring of some graphs

A proper edge coloring of a graph G is called adjacent vertex-distinguishing acyclic edge coloring if there is no 2-colored cycle in G and the coloring set of edges incident with u is not equal to the coloring set of edges incident with v, where uv ∈ E(G). The adjacent vertex distinguishing acyclic edge chromatic number of G, denoted by x′ _Aa (G), is the minimal number of colors in an adjacent vertex distinguishing acyclic edge coloring of G. If a graph G has an adjacent vertex distinguishing acyclic edge coloring, then G is called adjacent vertex distinguishing acyclic. In this paper, we obtain adjacent vertex-distinguishing acyclic edge coloring of some graphs and put forward some conjectures.     

Weighted coloring on planar, bipartite and split graphs: Complexity and approximation

We study complexity and approximation of min weighted node coloring in planar, bipartite and split graphs. We show that this problem is NP-hard in planar graphs, even if they are triangle-free and their maximum degree is bounded above by 4. Then, we prove that min weighted node coloring is NP-hard in P₈-free bipartite graphs, but polynomial for P₅-free bipartite graphs. We next focus on approximability in general bipartite graphs and improve earlier approximation results by giving approximation ratios matching inapproximability bounds. We next deal with min weighted edge coloring in bipartite graphs. We show that this problem remains strongly NP-hard, even in the case where the input graph is both cubic and planar. Furthermore, we provide an inapproximability bound of 7/6−ε, for any ε>0 and we give an approximation algorithm with the same ratio. Finally, we show that min weighted node coloring in split graphs can be solved by a polynomial time approximation scheme.     

Random coloring evolution on graphs

In this short note we study how two colors, red and blue, painted on a given graph are evolved randomly according to a transition rule which aims to simulate how people influence each other. We shall also calculate the probability that the evolution will be trapped eventually using the martingale method.     

Acyclic vertex coloring of graphs of maximum degree 5

An acyclic vertex coloring of a graph is a proper vertex coloring such that there are no bichromatic cycles. The acyclic chromatic number of G, denoted a(G), is the minimum number of colors required for acyclic vertex coloring of graph G. For a family F of graphs, the acyclic chromatic number of F, denoted by a(F), is defined as the maximum a(G) over all the graphs G∈F. In this paper we show that a(F)=8 where F is the family of graphs of maximum degree 5 and give a linear time algorithm to achieve this bound.     

Local coloring of Kneser graphs

A local coloring of a graph G is a function c:V(G)→N having the property that for each set S⊆V(G) with 2≤|S|≤3, there exist vertices u,v∈S such that |c(u)−c(v)|≥m_S, where m_S is the number of edges of the induced subgraph 〈S〉. The maximum color assigned by a local coloring c to a vertex of G is called the value of c and is denoted by χ_?(c). The local chromatic number of G is χ_?(G)=min{χ_?(c)}, where the minimum is taken over all local colorings c of G. The local coloring of graphs was introduced by Chartrand et al. [G. Chartrand, E. Salehi, P. Zhang, On local colorings of graphs, Congressus Numerantium 163 (2003) 207-221]. In this paper the local coloring of Kneser graphs is studied and the local chromatic number of the Kneser graph K(n,k) for some values of n and k is determined.