Isolated Toughness and Existence of f-Factors

Abstract

Let G be a graph with vertex set V(G) and edge set E(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; otherwise, set I(G) = |V(G)| − 1. Let f and g be two nonnegative integer-valued functions defined on V(G) satisfying a ≤ g(x) ≤ f(x) ≤ b . The purpose in this paper are to present sufficient conditions in terms of the isolated toughness and the minimum degree for graphs to have f-factors and (g, f)-factors (g < f). If g(x) ≡ a < b ≡ f(x), the conditions can be weakened.

This work is supported by Taishan Scholar Project of Shandong Province, NSFC of China (grant number 10471078), Natural Sciences and Engineering Research Council of Canada (grant number OGP0122059).