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     

Counterexamples to the List Square Coloring Conjecture

The square G² of a graph G is the graph defined on such that two vertices u and v are adjacent in G² if the distance between u and v in G is at most 2. Let and be the chromatic number and the list chromatic number of a graph H, respectively. A graph H is called chromatic‐choosable if . It is an interesting problem to find graphs that are chromatic‐choosable. Kostochka and Woodall (Choosability conjectures and multicircuits, Discrete Math., 240 (2001), 123–143) conjectured that for every graph G, which is called List Square Coloring Conjecture. In this article, we give infinitely many counter examples to the conjecture. Moreover, we show that the value can be arbitrarily large.     

On the circular chromatic number of circular partitionable graphs

This article studies the circular chromatic number of a class of circular partitionable graphs. We prove that an infinite family of circular partitionable graphs G has . A consequence of this result is that we obtain an infinite family of graphs G with the rare property that the deletion of each vertex decreases its circular chromatic number by exactly 1. © 2006 Wiley Periodicals, Inc. J Graph Theory     

χ‐bounded families of oriented graphs

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic number of a digraph is large enough, either the digraph contains a clique of size k or it contains S as an induced subgraph. As an evidence, we prove that for any oriented star S, every oriented graph with sufficiently large chromatic number contains either a transitive tournament of order 3 or S as an induced subdigraph. We then study for which sets of orientations of P₄ (the path on four vertices) similar statements hold. We establish some positive and negative results.     

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     

List colorings with measurable sets

The measurable list chromatic number of a graph G is the smallest number ξ such that if each vertex v of G is assigned a set L(v) of measure ξ in a fixed atomless measure space, then there exist sets such that each c(v) has measure one and for every pair of adjacent vertices v and v'. We provide a simpler proof of a measurable generalization of Hall's theorem due to Hilton and Johnson [J Graph Theory 54 (2007), 179–193] and show that the measurable list chromatic number of a finite graph G is equal to its fractional chromatic number. © 2008 Wiley Periodicals, Inc. J Graph Theory 59: 229–238, 2008     

Dimension,Graph and Hypergraph Coloring

There is a natural way to associate with a poset P a hypergraph H _P, called the hypergraph of incomparable pairs, so that the dimension of P is the chromatic number of H _P. The ordinary graph G _P of incomparable pairs determined by the edges in H _P of size 2 can have chromatic number substantially less than H _P. We give a new proof of the fact that the dimension of P is 2 if and only if G _P is bipartite. We also show that for each t 2, there exists a poset P _t for which the chromatic number of the graph of incomparable pairs of P _t is at most 3 t – 4, but the dimension of P _t is at least (3 / 2) ^{t – 1}. However, it is not known whether there is a function f: NN so that if P is a poset and the graph of incomparable pairs has chromatic number at most t, then the dimension of P is at most f(t).     

The distinguishing chromatic number of Cartesian products of two complete graphs

A labeling of a graph G is distinguishing if it is only preserved by the trivial automorphism of G. The distinguishing chromatic number of G is the smallest integer k such that G has a distinguishing labeling that is at the same time a proper vertex coloring. The distinguishing chromatic number of the Cartesian product is determined for all k and n. In most of the cases it is equal to the chromatic number, thus answering a question of Choi, Hartke and Kaul whether there are some other graphs for which this equality holds.     

Extremal problems for chromatic neighborhood sets

The chromatic neighborhood sequence of a graph G is the list of the chromatic numbers of the subgraphs induced by the neighborhoods of the vertices. We study the maximum multiplicity of this sequence, proving, amongst other things, that if a chromatic neighborhood sequence has t distinct values, the largest value being d_t, then there is a value with multiplicity at least . This bound is asymptotically tight. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 68–74, 2002     

Ohba’s conjecture for graphs with independence number five

Ohba has conjectured that if G is a k-chromatic graph with at most 2k+1 vertices, then the list chromatic number or choosability of G is equal to its chromatic number χ(G), which is k. It is known that this holds if G has independence number at most three. It is proved here that it holds if G has independence number at most five. In particular, and equivalently, it holds if G is a complete k-partite graph and each part has at most five vertices.     

Tree‐Chromatic Number Is Not Equal to Path‐Chromatic Number*

For a graph G and a tree‐decomposition of G, the chromatic number of is the maximum of , taken over all bags . The tree‐chromatic number of G is the minimum chromatic number of all tree‐decompositions of G. The path‐chromatic number of G is defined analogously. In this article, we introduce an operation that always increases the path‐chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path‐chromatic number and tree‐chromatic number are different. This settles a question of Seymour (J Combin Theory Ser B 116 (2016), 229–237). Our results also imply that the path‐chromatic numbers of the Mycielski graphs are unbounded.     

New bounds for the chromatic number of graphs

In this article we first give an upper bound for the chromatic number of a graph in terms of its degrees. This bound generalizes and modifies the bound given in 11 . Next, we obtain an upper bound of the order of magnitude for the coloring number of a graph with small K_2,t (as subgraph), where n is the order of the graph. Finally, we give some bounds for chromatic number in terms of girth and book size. These bounds improve the best known bound, in terms of order and girth, for the chromatic number of a graph when its girth is an even integer. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:110–122, 2008     

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)).     

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.     

Geometric graph homomorphisms

A geometric graph is a simple graph drawn on points in the plane, in general position, with straightline edges. A geometric homomorphism from to is a vertex map that preserves adjacencies and crossings. This work proves some basic properties of geometric homomorphisms and defines the geochromatic number as the minimum n so that there is a geometric homomorphism from to a geometric n‐clique. The geochromatic number is related to both the chromatic number and to the minimum number of plane layers of . By providing an infinite family of bipartite geometric graphs, each of which is constructed of two plane layers, which take on all possible values of geochromatic number, we show that these relationships do not determine the geochromatic number. This article also gives necessary (but not sufficient) and sufficient (but not necessary) conditions for a geometric graph to have geochromatic number at most four. As a corollary, we get precise criteria for a bipartite geometric graph to have geochromatic number at most four. This article also gives criteria for a geometric graph to be homomorphic to certain geometric realizations of K_{2, 2} and K_{3, 3}. © 2011 Wiley Periodicals, Inc. J Graph Theory 69:97‐113, 2012     

Finite fields and the 1‐chromatic number of orientable surfaces

The 1‐chromatic number χ₁(S_p) of the orientable surface S_p of genus p is the maximum chromatic number of all graphs which can be drawn on the surface so that each edge is crossed by no more than one other edge. We show that if there exists a finite field of order 4m+1, m≥3, then 8m+2≤χ₁(S)≤8m+3, where 8m+3 is Ringel's upper bound on χ₁(S). © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 179–184, 2010     

Nordhaus-Gaddum inequalities for the fractional and circular chromatic numbers

For a graph G on n vertices with chromatic number χ(G), the Nordhaus-Gaddum inequalities state that , and . Much analysis has been done to derive similar inequalities for other graph parameters, all of which are integer-valued. We determine here the optimal Nordhaus-Gaddum inequalities for the circular chromatic number and the fractional chromatic number, the first examples of Nordhaus-Gaddum inequalities where the graph parameters are rational-valued.     

On the Roots of σ‐Polynomials

Given a graph G of order n, the σ‐polynomial of G is the generating function where is the number of partitions of the vertex set of G into i nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729–756) proved that σ‐polynomials of graphs with chromatic number at least had all real roots, and conjectured the same held for chromatic number . We affirm this conjecture.