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In this paper we study the chromatic number of graphs with two prescribed induced cycle lengths. It is due to Sumner that triangle-free and P₅-free or triangle-free, P₆-free and C₆-free graphs are 3-colourable. A canonical extension of these graph classes is , the class of all graphs whose induced cycle lengths are 4 or 5. Our main result states that all graphs of are 3-colourable. Moreover, we present polynomial time algorithms to 3-colour all triangle-free graphs G of this kind, i.e., we have...
A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. We examine several subclasses of well-covered graphs of girth ≥ 4 with respect to the odd girth of the...
Let G = (V(G),E(G)) be a graph, and let k ≥ 1 be an integer. A set S ⊆ V(G) is called a global offensive k-alliance if |N(v)∩S| ≥ |N(v)-S|+k for every v ∈ V(G)-S, where N(v) is the neighborhood of v. The global offensive k-alliance number is the minimum cardinality of a global offensive k-alliance in G. We present different bounds on in terms of order, maximum degree, independence number, chromatic number and minimum degree.
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