Characterization of 1-tough graphs using factors

Abstract

For a graph G, let odd(G) and ω(G) denote the number of odd components and the number of components of G, respectively. Then it is well-known that G has a 1-factor if and only if odd(G−S)≤|S| for all S⊂V(G). Also it is clear that odd(G−S)≤ω(G−S). In this paper we characterize a 1-tough graph G, which satisfies ω(G−S)≤|S| for all 0̸≠S⊂V(G), using an H-factor of a set-valued function H:V(G)→{{1},{0,2}}. Moreover, we generalize this characterization to a graph that satisfies ω(G−S)≤f(S) for all 0̸≠S⊂V(G), where f:V(G)→{1,3,5,…}.