Closure and forbidden pairs for 2-factors

Abstract

Pairs of connected graphs X , Y such that a graph G is 2-connected and X Y -free implies that G is hamiltonian were characterized by Bedrossian. Using the closure concept for claw-free graphs, the first author simplified the characterization by showing that if considering the closure of G , the list in the Bedrossian characterization can be reduced to one pair, namely, K 1 , 3 , N 1 , 1 , 1 (where K i , j is the complete bipartite graph, and N i , j , k is the graph obtained by identifying endvertices of three disjoint paths of lengths i , j , k to the vertices of a triangle). Faudree et al. characterized pairs X , Y such that G is 2-connected and X Y -free implies that G has a 2-factor. Recently, the first author et al. strengthened the closure concept for claw-free graphs such that the closure of a graph has stronger properties while still preserving the (non)-existence of a 2-factor. In this paper we show that, using the 2-factor closure, the list of forbidden pairs for 2-factors can be reduced to two pairs, namely, K 1 , 4 , P 4 and K 1 , 3 , N 1 , 1 , 3 .