Open problems on graph coloring for special graph classes.

For a given graph G and integer k, the Coloring problem is that of testing whether G has a k-coloring, that is, whether there exists a vertex mapping c:V→{1,2,…}c:V→{1,2,…} such that c(u)≠c(v)c(u)≠c(v) for every edge uv∈Euv∈E. We survey known results on the computational complexity of Coloring for graph classes that are hereditary or for which some graph parameter is bounded. We also consider coloring variants, such as precoloring extensions and list colorings and give some open problems in the area of on-line coloring

A connection between circular colorings and periodic schedules

AbstractWe show that there is a curious connection between circular colorings of edge-weighted digraphs and periodic schedules of timed marked graphs. Circular coloring of an edge-weighted digraph was introduced by Mohar [B. Mohar, Circular colorings of edge-weighted graphs, J. Graph Theory 43 (2003) 107–116]. This kind of coloring is a very natural generalization of several well-known graph coloring problems including the usual circular coloring [X. Zhu, Circular chromatic number: A survey, Discrete Math. 229 (2001) 371–410] and the circular coloring of vertex-weighted graphs [W. Deuber, X. Zhu, Circular coloring of weighted graphs, J. Graph Theory 23 (1996) 365–376]. Timed marked graphs G→ [R.M. Karp, R.E. Miller, Properties of a model for parallel computations: Determinancy, termination, queuing, SIAM J. Appl. Math. 14 (1966) 1390–1411] are used, in computer science, to model the data movement in parallel computations, where a vertex represents a task, an arc uv with weight cuv represents a data channel with communication cost, and tokens on arc uv represent the input data of task vertex v. Dynamically, if vertex u operates at time t, then u removes one token from each of its in-arc; if uv is an out-arc of u, then at time t+cuv vertex u places one token on arc uv. Computer scientists are interested in designing, for each vertex u, a sequence of time instants {fu(1),fu(2),fu(3),…} such that vertex u starts its kth operation at time fu(k) and each in-arc of u contains at least one token at that time. The set of functions {fu:u∈V(G→)} is called a schedule of G→. Computer scientists are particularly interested in periodic schedules. Given a timed marked graph G→, they ask if there exist a period p>0 and real numbers xu such that G→ has a periodic schedule of the form fu(k)=xu+p(k−1) for each vertex u and any positive integer k. In this note we demonstrate an unexpected connection between circular colorings and periodic schedules. The aim of this note is to provide a possibility of translating problems and methods from one area of graph coloring to another area of computer science

An updated survey on rainbow connections of graphs - a dynamic survey

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow $k$-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study

A general framework for coloring problems: old results, new results, and open problems

In this survey paper we present a general framework for coloring problems that was introduced in a joint paper which the author presented at WG2003. We show how a number of different types of coloring problems, most of which have been motivated from frequency assignment, fit into this framework. We give a survey of the existing results, mainly based on and strongly biased by joint work of the author with several different groups of coauthors, include some new results, and discuss several open problems for each of the variants

Using Differential Evolution for the Graph Coloring

Differential evolution was developed for reliable and versatile function optimization. It has also become interesting for other domains because of its ease to use. In this paper, we posed the question of whether differential evolution can also be used by solving of the combinatorial optimization problems, and in particular, for the graph coloring problem. Therefore, a hybrid self-adaptive differential evolution algorithm for graph coloring was proposed that is comparable with the best heuristics for graph coloring today, i.e. Tabucol of Hertz and de Werra and the hybrid evolutionary algorithm of Galinier and Hao. We have focused on the graph 3-coloring. Therefore, the evolutionary algorithm with method SAW of Eiben et al., which achieved excellent results for this kind of graphs, was also incorporated into this study. The extensive experiments show that the differential evolution could become a competitive tool for the solving of graph coloring problem in the future

Vertex covers by monochromatic pieces - A survey of results and problems

This survey is devoted to problems and results concerning covering the vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles and other objects. It is an expanded version of the talk with the same title at the Seventh Cracow Conference on Graph Theory, held in Rytro in September 14-19, 2014.Comment: Discrete Mathematics, 201