Xuding Zhu

Circular chromatic number: a survey

Abstract The circular chromatic number χ c (G) of a graph G (also known as ‘the star-chromatic number’), is a natural generalization of the chromatic number of a graph. In this paper, we survey results on this topic, concentrating on the relations among the circular chromatic number, the chromatic number and some other parameters of a graph. Some of the results and/or proofs presented here are new. The last section is devoted to open problems. We pose 28 open problems, and discuss partial results and give references (if any) for each of these problems.

The Game Coloring Number of Planar Graphs

This paper discusses a variation of the game chromatic number of a graph: the game coloring number. This parameter provides an upper bound for the game chromatic number of a graph. We show that the game coloring number of a planar graph is at most 19. This implies that the game chromatic number of a planar graph is at most 19, which improves the previous known upper bound for the game chromatic number of planar graphs.

Duality and Polynomial Testing of Tree Homomorphisms

Let H be a fixed digraph. We consider the H-colouring problem, i.e., the problem of deciding which digraphs G admit a homomorphism to H. We are interested in a characterization in terms of the absence in G of certain tree-like obstructions. Specifically, we say that H has tree duality if, for all digraphs G, G is not homomorphic to H if and only if there is an oriented tree which is homomorphic to G but not to H. We prove that if H has tree duality then the H-colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the X-property studied by Gutjahr, Welzl, and Woeginger. We then focus on the case when H itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads H for which the H-colouring problem is NP -complete. We contrast these with several families of oriented triads H which have tree duality, or bounded treewidth duality, and hence polynomial H-colouring problems. If P 6= NP , then no oriented triad H with an NP -complete H-colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad H. We prove that none of the oriented triads H with NP -complete Hcolouring problems given in the companion paper has tree duality.

Star chromatic numbers and products of graphs

The star-chromatic number of a graph, a concept introduced by Vince, is natural generalization of the chromatic number of a graph. We point out an alternate definition of the star-chromatic number, which sheds new light on the relation of the star-chromatic number and the ordinary chromatic number. This new point of view allows us to answer several problems posed by Vince. We then study the starchromatic number from the perspective of graph homomorphisms and of graph products.

Multilevel Distance Labelings for Paths and Cycles

For a graph

Game chromatic number of outerplanar graphs

G

Acyclic and oriented chromatic numbers of graphs

, let

Refined activation strategy for the marking game

\diam(G)

A bound for the game chromatic number of graphs

denote the diameter of

The game coloring number of pseudo partial k -trees

G