A b-colouring of a graph G is a proper colouring of G such that each colour contains a vertex that is adjacent to all other colours and the b-chromatic number χb(G) is the maximum number of colours used in a b-colouring of G. If m(G) is the largest integer k such that G has at least k vertices with degree at least k − 1, then we know that χb(G) ≤ m(G). Irving and Manlove [1] prove that, if T is...

The circular chromatic number chi (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. T...

y Abstract We show that if a graph has acyclic chromatic number k, then its game chromatic number is at most k(k + 1). By applying the known upper bounds for the acyclic chromatic numbers of various classes of graphs, we obtain upper bounds for the game chromatic number of these classes of graphs. In particular, since a planar graph has acyclic chromatic number at most 5, we conclude that the g...

Let D be a digraph. The chromatic number χ(D) of D is the smallest number of colors needed to color the vertices of D such that every color class induces an acyclic subdigraph. The girth of D is the length of a shortest directed cycle, or ∞ if D is acyclic. Let G(k, n) be the maximum possible girth of a digraph on n vertices with χ(D) > k. It is shown that G(k, n) ≥ n1/k and G(k, n) ≤ (3 log2 n...