The r-inflation of a graph G is the lexicographic product G with Kr. A graph is said to have thickness t if its edges can be partitioned into t sets, each of which induces a planar graph, and t is smallest possible. In the setting of the r-inflation of planar graphs, we investigate the generalization of Ringel’s famous Earth-Moon problem: What is the largest chromatic number of any thickness-t ...

Let Z be the set of all integers and M a set of positive integers. The distance graph G(Z,M) generated by M is the graph with vertex set Z and in which i and j are adjacent whenever |i − j| ∈ M . Supported in part by the National Science Foundation under grant DMS 9805945. Supported in part by the National Science Council, R. O. C., under grant NSC892115-M-110-012.

It is shown that the difference between the chromatic number χ and the fractional chromatic number χf can be arbitrarily large in the class of uniquely colorable, vertex transitive graphs. For the lexicographic product G ◦ H it is shown that χ(G ◦ H) ≥ χf (G)χ(H). This bound has several consequences, in particular it unifies and extends several known lower bounds. Lower bounds of Stahl (for gen...

In this article we propose a new model for scheduling periodic tasks. The model is based on a variation of the circular chromatic number, called the multiple circular colouring of the conflict graph. We show that for a large class of graphs, this new model will provide better solutions than the original circular chromatic number. At the same time, it allows us to avoid the difficulty of impleme...

It is well known that for any k and g, there is a graph with chromatic number at least k and girth at least g. In 1970’s, Erdős and Hajnal conjectured that for any numbers k and g, there exists a number f(k, g), such that every graph with chromatic number at least f(k, g) contains a subgraph with chromatic number at least k and girth at least g. In 1978, Rödl proved the case for g = 4 and arbit...

the chromatic number of a graph g, denoted by χ(g), is the minimum number of colors such that g can be colored with these colors in such a way that no two adjacent vertices have the same color. a clique in a graph is a set of mutually adjacent vertices. the maximum size of a clique in a graph g is called the clique number of g. the turán graph tn(k) is a complete k-partite graph whose partition...

The local chromatic number is a coloring parameter defined as the minimum number of colors that should appear in the most colorful closed neighborhood of a vertex under any proper coloring of the graph. Its directed version is the same when we consider only outneighborhoods in a directed graph. For digraphs with all arcs being present in both directions the two values are obviously equal. Here ...