On the higher-order edge toughness of a graph

Abstract

For an integer c, 1≤c≤{curly logical or}V(G){curly logical or}-1, we define the cth-order edge toughness of a graph G as tc(G)=min |X| ω(G-X)-cX⊆E(G) & ω(G-X)>c The objective of this paper is to study this generalized concept of edge toughness. Besides giving the of the cth-order edge toughness τc(G) of a graph G, we prove that 'τc(G)≥k if and only if G has k edge-disjoint spanning forests with exactly c components'. We also study the 'balancity' of a graph G of order p and size q, which is defined as the smallest positive integer c such that τc(G) = p/(p-c). © 1993.