Isolated Toughness and Existence of [a, b]-factors in Graphs

For a graph G with vertex set V (G) and edge set E(G), let i(G) be the number of isolated vertices in G, the isolated toughness of G is defined as I(G) = min{|S|/i(G − S) | S ⊆ V (G), i(G − S) ≥ 2}, if G is not complete; and I(Kn) = n − 1. In this paper, we investigate the existence of [a, b]-factor in terms of this graph invariant. We proved that if G is a graph with δ(G) ≥ a and I(G) ≥ a, then G has a fractional a-factor. Moreover, if δ(G) ≥ a, I(G)> (a−1)+ a−1b and G−S has no (a−1)-regular component for any subset S of V (G), then G has an [a, b]-factor. The later result is a generalization of Katerinis’ well-known theorem about [a, b]-factors (P. Katerinis, Toughness of graphs and the existence of factors, Discrete Math. 80(1990), 81-92)