A new upper bound for the harmonious chromatic number

A harmonious coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number h(G) is the least number of colors in such a coloring. We obtain a new upper bound for the harmonious chromatic number of general graphs. © 1998 John Wiley & Sons, Inc. J Graph Theory 29: 257–261, 1998     

Achromatic and Harmonious Colorings of Circulant Graphs

A proper vertex coloring of a graph G is achromatic (respectively harmonious) if every two colors appear together on at least one (resp. at most one) edge. The largest (resp. the smallest) number of colors in an achromatic (resp. a harmonious) coloring of G is called the achromatic (resp. harmonious chromatic) number of G and denoted by (resp. ). For a finite set of positive integers and a positive integer n, a circulant graph, denoted by , is an undirected graph on the set of vertices that has an edge if and only if either or is a member of (where substraction is computed modulo n). For any fixed set , we show that is asymptotically equal to , with the error term . We also prove that is asymptotically equal to , with the error term . As corollaries, we get results that improve, for a fixed k, the previously best estimations on the lengths of a shortest k‐radius sequence over an n‐ary alphabet (i.e., a sequence in which any two distinct elements of the alphabet occur within distance k of each other) and a longest packing k‐radius sequence over an n‐ary alphabet (which is a dual counterpart of a k‐radius sequence).     

The Pseudoachromatic Number of a Graph

The pseudoachromatic number of a graph G is the maximum size of a vertex partition of G (where the sets of the partition may or may not be independent) such that, between any two distinct parts, there is at least one edge of G. This parameter is determined for graphs such as cycles, paths, wheels, certain complete multipartite graphs, and for other classes of graphs. Some open problems are raised.AMS Subject Classification (1991): primary 05C75 secondary 05C85     

Approximation Algorithms for the Achromatic Number

The achromatic number for a graph G = V, E is the largest integer m such that there is a partition of V into disjoint independent sets {V₁, …, V_m} such that for each pair of distinct sets V_i, V_j, V_i V_j is not an independent set in G. Yannakakis and Gavril (1980, SIAM J. Appl. Math.38, 364–372) proved that determining this value for general graphs is NP-complete. For n-vertex graphs we present the first o(n) approximation algorithm for this problem. We also present an O(n^5/12) approximation algorithm for graphs with girth at least 5 and a constant approximation algorithm for trees.     

On the complexity of the circular chromatic number

Circular chromatic number, χ_c is a natural generalization of chromatic number. It is known that it is NP ‐hard to determine whether or not an arbitrary graph G satisfies χ(G)=χ_c(G). In this paper we prove that this problem is NP ‐hard even if the chromatic number of the graph is known. This answers a question of Xuding Zhu. Also we prove that for all positive integers k ≥ 2 and n ≥ 3, for a given graph G with χ(G) = n, it is NP ‐complete to verify if . © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 226–230, 2004     

Achromatic number of fragmentable graphs

A complete coloring of a simple graph G is a proper vertex coloring such that each pair of colors appears together on at least one edge. The achromatic number ψ(G) is the greatest number of colors in such a coloring. We say a class of graphs is fragmentable if for any positive ε, there is a constant C such that any graph in the class can be broken into pieces of size at most C by removing a proportion at most ε of the vertices. Examples include planar graphs and grids of fixed dimension. Determining the achromatic number of a graph is NP‐complete in general, even for trees, and the achromatic number is known precisely for only very restricted classes of graphs. We extend these classes very considerably, by giving, for graphs in any class which is fragmentable, triangle‐free, and of bounded degree, a necessary and sufficient condition for a sufficiently large graph to have a complete coloring with a given number of colors. For the same classes, this gives a tight lower bound for the achromatic number of sufficiently large graphs, and shows that the achromatic number can be determined in polynomial time. As examples, we give exact values of the achromatic number for several graph families. © 2009 Wiley Periodicals, Inc. J Graph Theory 65:94–114, 2010     

About the upper chromatic number of a co-hypergraph

A mixed hypergraph consists of two families of subsets: the edges and the co-edges. In a coloring every co-edge has at least two vertices of the same color, and every edge has at least two vertices of different colors. The largest and smallest possible number of colors in a coloring is termed the upper and lower chromatic numbers, respectively. In this paper we investigate co-hypergraphs i.e., the hypergraphs with only co-edges, with respect to the property of coloring. The relationship between the lower bound of the size of co-edges and the lower bound of the upper chromatic number is explored. The necessary and sufficient conditions for determining the upper chromatic numbers, of a co-hypergraph are provided. And the bounds of the number of co-edges of some uniform co-hypergraphs with certain upper chromatic numbers are given.     

The circular chromatic number of the Mycielskian of G

In a search for triangle-free graphs with arbitrarily large chromatic numbers, Mycielski developed a graph transformation that transforms a graph G into a new graph μ(G), we now call the Mycielskian of G, which has the same clique number as G and whose chromatic number equals χ(G) + 1. Chang, Huang, and Zhu [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear] have investigated circular chromatic numbers of Mycielskians for several classes of graphs. In this article, we study circular chromatic numbers of Mycielskians for another class of graphs G. The main result is that χ_c(μ(G)) = χ(μ(G)), which settles a problem raised in [G. J. Chang, L. Huang, & X. Zhu, Discrete Math, to appear, and X. Zhu, to appear]. As χ_c(G) = and χ(G) = , consequently, there exist graphs G such that χ_c(G) is as close to χ(G) − 1 as you want, but χ_c(μ(G)) = χ(μ(G)). © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 63–71, 1999     

Circular chromatic number and Mycielski construction

This paper gives a sufficient condition for a graph G to have its circular chromatic number equal to its chromatic number. By using this result, we prove that for any integer t ≥ 1, there exists an integer n such that for all . © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 106–115, 2003     

A note on the least number of edges of 3-uniform hypergraphs with upper chromatic number 2

The upper chromatic number of a hypergraph H=(X,E) is the maximum number k for which there exists a partition of X into non-empty subsets X=X₁∪X₂∪?∪X_k such that for each edge at least two vertices lie in one of the partite sets. We prove that for every n?3 there exists a 3-uniform hypergraph with n vertices, upper chromatic number 2 and ⌈n(n-2)/3⌉ edges which implies that a corresponding bound proved in [K. Diao, P. Zhao, H. Zhou, About the upper chromatic number of a co-hypergraph, Discrete Math. 220 (2000) 67-73] is best-possible.     

Hadwiger number and chromatic number for near regular degree sequences

We consider a problem related to Hadwiger's Conjecture. Let D=(d₁, d₂, …, d_n) be a graphic sequence with 0?d₁?d₂?···?d_n?n?1. Any simple graph G with D its degree sequence is called a realization of D. Let R[D] denote the set of all realizations of D. Define h(D)=max{h(G): G∈R[D]} and χ(D)=max{χ(G): G∈R[D]}, where h(G) and χ(G) are Hadwiger number and chromatic number of a graph G, respectively. Hadwiger's Conjecture implies that h(D)?χ(D). In this paper, we establish the above inequality for near regular degree sequences. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 175–183, 2010     

Edge‐face chromatic number and edge chromatic number of simple plane graphs

Given a simple plane graph G, an edge‐face k‐coloring of G is a function ? : E(G) ∪ F(G) → {1,…,k} such that, for any two adjacent or incident elements a, b ∈ E(G) ∪ F(G), ?(a) ≠ ?(b). Let χ_e(G), χ_ef(G), and Δ(G) denote the edge chromatic number, the edge‐face chromatic number, and the maximum degree of G, respectively. In this paper, we prove that χ_ef(G) = χ_e(G) = Δ(G) for any 2‐connected simple plane graph G with Δ (G) ≥ 24. © 2005 Wiley Periodicals, Inc. J Graph Theory     

On the complete chromatic number of Halin graphs

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.Write.1.IntroductionDefinition1.FOrany3-connectedplanargraphG(V,E,F)withA(G)23,iftheboundaryedgesoffacefowhichisadjacenttotheothersareremoved,itbecomesatree,andthedegreeofeachvertexofV(fo)is3,andthenGiscalledaHalingraph;foiscalledtheouterfaceofG,andtheotherscalledtheinteriorfaces,thevenicesonthefacefoarecalledtheoutervenices,theoillersarecalledtheinterior...ti..,tll.ForanyplanargraphG(V,E,F),f,f'eF,fisadjacenttof'ifan…     

The 6-relaxed game chromatic number of outerplanar graphs

Suppose G is a graph and k,d are integers. The (k,d)-relaxed colouring game on G is a game played by two players, Alice and Bob, who take turns colouring the vertices of G with legal colours from a set X of k colours. Here a colour i is legal for an uncoloured vertex x if after colouring x with colour i, the subgraph induced by vertices of colour i has maximum degree at most d. Alice’s goal is to have all the vertices coloured, and Bob’s goal is the opposite: to have an uncoloured vertex without a legal colour. The d-relaxed game chromatic number of G, denoted by , is the least number k so that when playing the (k,d)-relaxed colouring game on G, Alice has a winning strategy. This paper proves that if G is an outerplanar graph, then for d≥6.     

On the chromatic number of random graphs

We consider random graphs G_n,p with fixed edge-probability p. We refine an argument of Bollobás to show that almost all such graphs have chromatic number equal to n/{2 log_b n ? 2 log_b log_b n + O(1)} where b = 1/(1 ? p).     

Circular game chromatic number of graphs

In a circular r-colouring game on G, Alice and Bob take turns colouring the vertices of G with colours from the circle S(r) of perimeter r. Colours assigned to adjacent vertices need to have distance at least 1 in S(r). Alice wins the game if all vertices are coloured, and Bob wins the game if some uncoloured vertices have no legal colour. The circular game chromatic number χ_cg(G) of G is the infimum of those real numbers r for which Alice has a winning strategy in the circular r-colouring game on G. This paper proves that for any graph G, , where is the game colouring number of G. This upper bound is shown to be sharp for forests. It is also shown that for any graph G, χ_cg(G)≤2χ_a(G)(χ_a(G)+1), where χ_a(G) is the acyclic chromatic number of G. We also determine the exact value of the circular game chromatic number of some special graphs, including complete graphs, paths, and cycles.     

A bound on the chromatic number using the longest odd cycle length

It is known that for every integer k ≥ 4, each k‐map graph with n vertices has at most kn ? 2k edges. Previously, it was open whether this bound is tight or not. We show that this bound is tight for k = 4, 5. We also show that this bound is not tight for large enough k (namely, k ≥ 374); more precisely, we show that for every and for every integer , each k‐map graph with n vertices has at most edges. This result implies the first polynomial (indeed linear) time algorithm for coloring a given k‐map graph with less than 2k colors for large enough k. We further show that for every positive multiple k of 6, there are infinitely many integers n such that some k‐map graph with n vertices has at least edges. © 2007 Wiley Periodicals, Inc. J Graph Theory 55: 267–290, 2007