Forbidden triples generating a finite set of graphs with minimum degree three

Abstract

For a family H of graphs, a graph G is said to be H -free if G contains no member of H as an induced subgraph. We let G ̃ 3 ( H ) denote the family of connected H -free graphs having minimum degree at least 3. In this paper, we characterize the families H of connected graphs with | H | = 3 such that G ̃ 3 ( H ) is a finite family, except for the case where { K 3 , K 2 , 2 } ⊆ H . Our results give a fast algorithm determining the chromatic number of several classes of graphs with a forbidden subgraph condition imposed on.