We prove several results about three families of graphs. For queen graphs, defined from the usual moves of a chess queen, we find the edge-chromatic number in almost all cases. In the unproved case, we have a conjecture supported by a vast amount of computation, which involved the development of a new edge-coloring algorithm. The conjecture is that the edge-chromatic number is the maximum degre...

The game chromatic number χg is considered for the Cartesian product G 2 H of two graphs G and H. We determine exact values of χg(G2H) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χg(G2H) is not bounded from above by a function of game chromatic numbers of graphs G and H. An analogous result is proved for the game coloring number colg(G2...

We develop a new upper bound, called the nested chromatic number, for the chromatic number of a finite simple graph. This new invariant can be computed in polynomial time, unlike the standard chromatic number which is NP -hard. We further develop multiple distinct bounds on the nested chromatic number using common properties of graphs. We also determine the behavior of the nested chromatic numb...

An injective coloring of graphs is a vertex coloring where two vertices receive distinct colors if they have a common neighbor. In 1977, Wegner [12] posed a conjecture on the chromatic number of squares of graphs which remains unsolved for planar graphs with maximum degrees ∆ ≥ 4. Obviously, the chromatic number of the square of a graph is at least the injective chromatic number of a graph. We ...

In this paper we obtain some upper bounds for b-chromatic number of K1,t -free graphs, graphs with given minimum clique partition and bipartite graphs. These bounds are in terms of either clique number or chromatic number of graphs or biclique number for bipartite graphs. We show that all the bounds are tight. AMS Classification: 05C15.

First László Székely and more recently Saharon Shelah and Alexander Soifer have presented examples of infinite graphs whose chromatic numbers depend on the axioms chosen for set theory. The existence of such graphs may be relevant to the Chromatic Number of the Plane problem. In this paper we construct a new class of graphs with ambiguous chromatic number. They are unit distance graphs with ver...

where α(X) is the independence number of the subgraph of G induced by X. The independence ratio is a relaxation of the chromatic number χ(G) in the sense that χ(G) ≥ ι(G) for every graph G, while for many natural classes of graphs these quantities are almost equal. In this paper, we address two old conjectures of Erdős on cycles in graphs with large chromatic number and a conjecture of Erdős an...