Tight Toughness, Isolated Toughness and Binding Number Bounds for the $$\{K_2,C_n\}$$-Factors

Toughness and binding number

Let τ(G) and bind(G) be the toughness and binding number, respectively, of a graph G. Woodall observed in 1973 that τ(G) > bind(G) − 1. In this paper we obtain best possible improvements of this inequality except when (1+ √ 5)/2 < bind(G) < 2 and bind(G) has even denominator when expressed in lowest terms.

Isolated Toughness and Existence of f -Factors

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 edge-toughness

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 ...

Toughness, degrees and 2-factors

In this paper we generalize a Theorem of Jung which shows that 1-tough graphs with (G) |V (G)|−4 2 are hamiltonian. Our generalization shows that these graphs contain a wide variety of 2-factors. In fact, these graphs contain not only 2-factors having just one cycle (the hamiltonian case) but 2-factors with k cycles, for any k such that 1 k n−16 4 . © 2004 Elsevier B.V. All rights reserved.