A new coloring theorem of Kneser graphs

In 1997, Johnson, Holroyd and Stahl conjectured that the circular chromatic number of the Kneser graphs KG(n,k) is equal to the chromatic number of these graphs. This was proved by Simonyi and Tardos (2006) [13] and independently by Meunier (2005) [10], if χ(KG(n,k)) is even. In this paper, we propose an alternative version of Kneser's coloring theorem to confirm the Johnson-Holroyd-Stahl conjecture.     

On the locating chromatic number of Kneser graphs

Let c be a proper k-coloring of a connected graph G and Π=(C₁,C₂,…,C_k) be an ordered partition of V(G) into the resulting color classes. For a vertex v of G, the color code of v with respect to Π is defined to be the ordered k-tuple c_Π(v):=(d(v,C₁),d(v,C₂),…,d(v,C_k)), where d(v,C_i)=min{d(v,x)|x∈C_i},1≤i≤k. If distinct vertices have distinct color codes, then c is called a locating coloring. The minimum number of colors needed in a locating coloring of G is the locating chromatic number of G, denoted by χ_L(G). In this paper, we study the locating chromatic number of Kneser graphs. First, among some other results, we show that χ_L(KG(n,2))=n−1 for all n≥5. Then, we prove that χ_L(KG(n,k))≤n−1, when n≥k². Moreover, we present some bounds for the locating chromatic number of odd graphs.     

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     

Multichromatic numbers,star chromatic numbers and Kneser graphs

We investigate the relation between the multichromatic number (discussed by Stahl and by Hilton, Rado and Scott) and the star chromatic number (introduced by Vince) of a graph. Denoting these by χ* and η*, the work of the above authors shows that χ*(G) = η*(G) if G is bipartite, an odd cycle or a complete graph. We show that χ*(G) ≤ η*(G) for any finite simple graph G. We consider the Kneser graphs , for which χ* = m/n and η*(G)/χ*(G) is unbounded above. We investigate particular classes of these graphs and show that η* = 3 and η* = 4; (n ≥ 1), and η* = m - 2; (m ≥ 4). © 1997 John Wiley & Sons, Inc. J Graph Theory 26: 137–145, 1997     

On the diameter of generalized Kneser graphs

Let r, k be positive integers, s(<r), a nonnegative integer, and n=2r-s+k. The set of r-subsets of [n]={1,2,…,n} is denoted by [n]^r. The generalized Kneser graph K(n,r,s) is the graph whose vertex-set is [n]^r where two r-subsets A and B are joined by an edge if |A∩B|?s. This note determines the diameter of generalized Kneser graphs. More precisely, the diameter of K(n,r,s) is equal to , which generalizes a result of Valencia-Pabon and Vera [On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

Clique coloring of binomial random graphs

A clique coloring of a graph is a coloring of the vertices so that no maximal clique is monochromatic (ignoring isolated vertices). The smallest number of colors in such a coloring is the clique chromatic number. In this paper, we study the asymptotic behavior of the clique chromatic number of the random graph ??(n,p) for a wide range of edge‐probabilities p = p(n). We see that the typical clique chromatic number, as a function of the average degree, forms an intriguing step function.     

The harmonious coloring problem is NP-complete for interval and permutation graphs

In this paper, we prove that the harmonious coloring problem is NP-complete for connected interval and permutation graphs. Given a simple graph G, a harmonious coloring of G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number is the least integer k for which G admits a harmonious coloring with k colors. Extending previous work on the NP-completeness of the harmonious coloring problem when restricted to the class of disconnected graphs which are simultaneously cographs and interval graphs, we prove that the problem is also NP-complete for connected interval and permutation graphs.     

Acyclic edge coloring of subcubic 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 it is denoted by a^′(G). From a result of Burnstein it follows that all subcubic graphs are acyclically edge colorable using five colors. This result is tight since there are 3-regular graphs which require five colors. In this paper we prove that any non-regular connected graph of maximum degree 3 is acyclically edge colorable using at most four colors. This result is tight since all edge maximal non-regular connected graphs of maximum degree 3 require four colors.     

Circular coloring of signed graphs

Let ( be two positive integers. We generalize the well‐studied notions of ‐colorings and of the circular chromatic number to signed graphs. This implies a new notion of colorings of signed graphs, and the corresponding chromatic number χ. Some basic facts on circular colorings of signed graphs and on the circular chromatic number are proved, and differences to the results on unsigned graphs are analyzed. In particular, we show that the difference between the circular chromatic number and the chromatic number of a signed graph is at most 1. Indeed, there are signed graphs where the difference is 1. On the other hand, for a signed graph on n vertices, if the difference is smaller than 1, then there exists , such that the difference is at most . We also show that the notion of ‐colorings is equivalent to r‐colorings (see [12] (X. Zhu, Recent developments in circular coloring of graphs, in Topics in Discrete Mathematics Algorithms and Combinatorics Volume 26 , Springer Berlin Heidelberg, 2006, pp. 497–550)).     

Semistrong edge coloring of graphs

Weakening the notion of a strong (induced) matching of graphs, in this paper, we introduce the notion of a semistrong matching. A matching M of a graph G is called semistrong if each edge of M has a vertex, which is of degree one in the induced subgraph G[M]. We strengthen earlier results by showing that for the subset graphs and for the Kneser graphs the sizes of the maxima of the strong and semistrong matchings are equal and so are the strong and semistrong chromatic indices. Similar properties are conjectured for the n‐dimensional cube. © 2005 Wiley Periodicals, Inc. J Graph Theory 49: 39–47, 2005     

On adjacent-vertex-distinguishing total coloring of graphs

In this paper, we present a new concept of the adjacent-vertex-distinguishing total coloring of graphs (briefly, AVDTC of graphs) and, meanwhile, have obtained the adjacent-vertex-distinguishing total chromatic number of some graphs such as cycle, complete graph, complete bipartite graph, fan, wheel and tree.     

Non-cover generalized Mycielski, Kneser, and Schrijver graphs

A graph is said to be a cover graph if it is the underlying graph of the Hasse diagram of a finite partially ordered set. We prove that the generalized Mycielski graphs M_m(C_2t+1) of an odd cycle, Kneser graphs KG(n,k), and Schrijver graphs SG(n,k) are not cover graphs when m?0,t?1, k?1, and n?2k+2. These results have consequences in circular chromatic number.     

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     

List coloring of graphs having cycles of length divisible by a given number

Let G be a graph and χ_l(G) denote the list chromatic number of G. In this paper we prove that for every graph G for which the length of each cycle is divisible by l (l≥3), χ_l(G)≤3.     

Game coloring the Cartesian product of graphs

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G^* be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G^* has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G□G′ has game chromatic number at most k(k + m ? 1). As a consequence, the Cartesian product of two forests has game chromatic number at most 10, and the Cartesian product of two planar graphs has game chromatic number at most 105. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 261–278, 2008