Circular choosability is rational

The circular choosability or circular list chromatic number of a graph is a list-version of the circular chromatic number, that was introduced by Mohar in 2002 and has been studied by several groups of authors since then. One of the nice properties that the circular chromatic number enjoys is that it is a rational number for all finite graphs G, and a fundamental question, posed by Zhu and reiterated by others, is whether the same holds for the circular choosability. In this paper we show that this is indeed the case.     

Distinguishing chromatic numbers of complements of Cartesian products of complete graphs

The distinguishing chromatic number of a graph, $G$, is the minimum number of colours required to properly colour the vertices of $G$ so that the only automorphism of $G$ that preserves colours is the identity. There are many classes of graphs for which the distinguishing chromatic number has been studied, including Cartesian products of complete graphs (Jerebic and Klav?ar, 2010). In this paper we determine the distinguishing chromatic number of the complement of the Cartesian product of complete graphs, providing an interesting class of graphs, some of which have distinguishing chromatic number equal to the chromatic number, and others for which the difference between the distinguishing chromatic number and chromatic number can be arbitrarily large.     

Fractional total colourings of graphs of high girth

Reed conjectured that for every ?>0 and Δ there exists g such that the fractional total chromatic number of a graph with maximum degree Δ and girth at least g is at most Δ+1+?. We prove the conjecture for Δ=3 and for even Δ?4 in the following stronger form: For each of these values of Δ, there exists g such that the fractional total chromatic number of any graph with maximum degree Δ and girth at least g is equal to Δ+1.     

一类循环图中的Eulerian定向数

一般的图中Eulerian定向数的计数是#P-完全问题,但对于某些特殊图中的Eulerian定向数给出精确计数是完全有可能的.通过拆分解构的方法可以找到与一类循环图中Eulerian定向数有关的递推关系,从而给出该数的精确计数.前人的工作在于给出了一些近似估计.     

Enumerating maximal independent sets with applications to graph colouring

We give tight upper bounds on the number of maximal independent sets of size k (and at least k and at most k) in graphs with n vertices. As an application of the proof, we construct improved algorithms for graph colouring and computing the chromatic number of a graph.     

Local Chromatic Number, KY Fan's Theorem, And Circular Colorings

The local chromatic number of a graph was introduced in [14]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [17] which generalizes the Borsuk–Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K_{⌈t/2⌉,⌊t/2⌋} in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [18].) This yields a lower bound of ⌈t/2⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier [42]. We also show that odd chromatic Schrijver graphs behave differently, their circular chromatic number can be arbitrarily close to the other extreme. * Research partially supported by the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046376, AT048826, and NK62321. † Research partially supported by the NSERC grant 611470 and the Hungarian Foundation for Scientific Research Grant (OTKA) Nos. T037846, T046234, AT048826, and NK62321.     

Edge coloring nearly bipartite graphs

We give a simple polynomial time algorithm to compute the chromatic index of graphs which can be made bipartite by deleting a vertex. An analysis of this algorithm shows that for such graphs, the chromatic index is the roundup of the fractional chromatic index.     

On the chromatic number of circulant graphs

Given a set D of a cyclic group C, we study the chromatic number of the circulant graph G(C,D) whose vertex set is C, and there is an edge ij whenever i−j∈D∪−D. For a fixed set D={a,b,c:a<b<c} of positive integers, we compute the chromatic number of circulant graphs G(Z_N,D) for all N≥4bc. We also show that, if there is a total order of D such that the greatest common divisors of the initial segments form a decreasing sequence, then the chromatic number of G(Z,D) is at most 4. In particular, the chromatic number of a circulant graph on Z_N with respect to a minimum generating set D is at most 4. The results are based on the study of the so-called regular chromatic number, an easier parameter to compute. The paper also surveys known results on the chromatic number of circulant graphs.     

Acyclic improper colouring of graphs with maximum degree 4

A k-colouring(not necessarily proper) of vertices of a graph is called acyclic, if for every pair of distinct colours i and j the subgraph induced by the edges whose endpoints have colours i and j is acyclic. We consider acyclic k-colourings such that each colour class induces a graph with a given(hereditary) property. In particular, we consider acyclic k-colourings in which each colour class induces a graph with maximum degree at most t, which are referred to as acyclic t-improper k-colourings. The acyclic t-improper chromatic number of a graph G is the smallest k for which there exists an acyclic t-improper k-colouring of G. We focus on acyclic colourings of graphs with maximum degree 4. We prove that 3 is an upper bound for the acyclic 3-improper chromatic number of this class of graphs. We also provide a non-trivial family of graphs with maximum degree4 whose acyclic 3-improper chromatic number is at most 2, namely, the graphs with maximum average degree at most 3. Finally, we prove that any graph G with Δ(G) 4 can be acyclically coloured with 4 colours in such a way that each colour class induces an acyclic graph with maximum degree at most 3.     

The fractional chromatic number of infinite graphs

The fractional chromatic number of a graph G is the infimum of the total weight that can be assigned to the independent sets of G in such a way that, for each vertex v of G, the sum of the weights of the independent sets containing v is at least 1. In this note we give a graph a graph whose fractional chromatic number is strictly greater than the supremum of the fractional chromatic numbers of its finite subgraphs. This answers a question of Zhu. We also give some grphs for which the fractional chromatic number is not attined, answering another of Zhu. © 1995 John Wiley & Sons, Inc.     

Acyclic improper colourings of graphs with bounded maximum degree

For graphs of bounded maximum degree, we consider acyclic t-improper colourings, that is, colourings in which each bipartite subgraph consisting of the edges between two colour classes is acyclic, and each colour class induces a graph with maximum degree at most t.We consider the supremum, over all graphs of maximum degree at most d, of the acyclic t-improper chromatic number and provide t-improper analogues of results by Alon, McDiarmid and Reed [N. Alon, C.J.H. McDiarmid, B. Reed, Acyclic coloring of graphs, Random Structures Algorithms 2 (3) (1991) 277-288] and Fertin, Raspaud and Reed [G. Fertin, A. Raspaud, B. Reed, Star coloring of graphs, J. Graph Theory 47 (3) (2004) 163-182].     

Fractional chromatic number and circular chromatic number for distance graphs with large clique size

An Erratum has been published for this article in Journal of Graph Theory 48: 329–330, 2005 . Let M be a set of positive integers. The distance graph generated by M, denoted by G(Z, M), has the set Z of all integers as the vertex set, and edges ij whenever |i?j| ∈ M. We investigate the fractional chromatic number and the circular chromatic number for distance graphs, and discuss their close connections with some number theory problems. In particular, we determine the fractional chromatic number and the circular chromatic number for all distance graphs G(Z, M) with clique size at least |M|, except for one case of such graphs. For the exceptional case, a lower bound for the fractional chromatic number and an upper bound for the circular chromatic number are presented; these bounds are sharp enough to determine the chromatic number for such graphs. Our results confirm a conjecture of Rabinowitz and Proulx 22 on the density of integral sets with missing differences, and generalize some known results on the circular chromatic number of distance graphs and the parameter involved in the Wills' conjecture 26 (also known as the “lonely runner conjecture” 1 ). © 2004 Wiley Periodicals, Inc. J Graph Theory 47: 129–146, 2004     

Graphes cubiques d'indice trois,graphes cubiques isochromatiques,graphes cubiques d'indice quatre

The process introduced by E. Johnson [Amer. Math. Monthly73 (1966), 52–55] for constructing connected cubic graphs can be modified so as to obtain restricted classes of cubic graphs, in particular, those defined by their chromatic number or their chromatic index. We construct here the graphs of chromatic number three and the graphs whose chromatic number is equal to its chromatic index (isochromatic graphs). We then give results about the construction of the class of graphs of chromatic index four, and in particular, we construct an infinite class of “snarks.”     

Thickness-Two Graphs Part Two: More New Nine-Critical Graphs,Independence Ratio,Cloned Planar Graphs,and Singly and Doubly Outerplanar Graphs

There are numerous means for measuring the closeness to planarity of a graph such as crossing number, splitting number, and a variety of thickness parameters. We focus on the classical concept of the thickness of a graph, and we add to earlier work in [4]. In particular, we offer new 9-critical thickness-two graphs on 17, 25, and 33 vertices, all of which provide counterexamples to a conjecture on independence ratio of Albertson; we investigate three classes of graphs, namely singly and doubly outerplanar graphs, and cloned planar graphs. We give a sharp upper bound for the largest chromatic number of the cloned planar graphs, and we give upper and lower bounds for the largest chromatic number of the former two classes.     

Coloring graphs by translates in the circle

The fractional and circular chromatic numbers are the two most studied non-integral refinements of the chromatic number of a graph. Starting from the definition of a coloring base of a graph, which originated in work related to ergodic theory, we formalize the notion of a gyrocoloring of a graph: the vertices are colored by translates of a single Borel set in the circle group, and neighboring vertices receive disjoint translates. The corresponding gyrochromatic number of a graph always lies between the fractional chromatic number and the circular chromatic number. We investigate basic properties of gyrocolorings. In particular, we construct examples of graphs whose gyrochromatic number is strictly between the fractional chromatic number and the circular chromatic number. We also establish several equivalent definitions of the gyrochromatic number, including a version involving all finite abelian groups.     

Proof of a conjecture on fractional Ramsey numbers

Jacobson, Levin, and Scheinerman introduced the fractional Ramsey function r_f (a₁, a₂, …, a_k) as an extension of the classical definition for Ramsey numbers. They determined an exact formula for the fractional Ramsey function for the case k=2. In this article, we answer an open problem by determining an explicit formula for the general case k>2 by constructing an infinite family of circulant graphs for which the independence numbers can be computed explicitly. This construction gives us two further results: a new (infinite) family of star extremal graphs which are a superset of many of the families currently known in the literature, and a broad generalization of known results on the chromatic number of integer distance graphs. © 2009 Wiley Periodicals, Inc. J Graph Theory 63: 164–178, 2010     

Halin图的点着色算法

本文解决了Halin图的点色数问题,并给出了一个可在线性时间内对Halin图进行点着色的算法。     

Lower Bounds for Measurable Chromatic Numbers

The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinite program into an infinite linear program which then turns out to be an extremal question about Jacobi polynomials which we solve explicitly in the limit. As an application we derive new lower bounds for the measurable chromatic number of the Euclidean space in dimensions 10, . . . , 24 and we give a new proof that it grows exponentially with the dimension.     

2k-系整数可重集伴随等价图的计数

给出了由2k-系的整数组成的可重集的伴随等价图的个数问题,同时也给出了其补图的色等价图的个数.     

Improper colouring of (random) unit disk graphs

For any graph G, the k-improper chromatic numberχ^k(G) is the smallest number of colours used in a colouring of G such that each colour class induces a subgraph of maximum degree k. We investigate χ^k for unit disk graphs and random unit disk graphs to generalise results of McDiarmid and Reed [Colouring proximity graphs in the plane, Discrete Math. 199(1-3) (1999) 123-137], McDiarmid [Random channel assignment in the plane, Random Structures Algorithms 22(2) (2003) 187-212], and McDiarmid and Müller [On the chromatic number of random geometric graphs, submitted for publication].