A Note on Total Excess of Spanning Trees

Abstract

A graph G is said to be t-tough if |S| ≥ t · ω(G - S) for any subset S of V(G) with ω(G - S) ≥ 2, where ω(G - S) is the number of components in G - S. Win proved that for any integer n ≥ 3 every -tough graph has a spanning tree with maximum degree at most n. In this paper, we investigate t-tough graphs including the cases where t ∉ {1, ½, ⅓,…}, and consider spanning trees in such graphs. Using the notion of total excess, we prove that if G is -tough for an integer n ≥ 2 and a real number ϵ with , then G has a spanning tree T such thatWe also investigate the relation between spanning trees in a graph obtained by different pairs of parameters (n, ϵ). As a consequence, we prove the existence of “a universal tree” in a connected t-tough graph G, that is a spanning tree T such that ∑v∊V(T) max{0, degT(v) - n} ≤ ϵ|V(G)| - 2 for any integer n ≥ 2 and real number ϵ with , which satisfy t ≥ .