A Note on Total Excess of Spanning Trees

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 1 n−2 -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, 1 2 , 1 3 , . . .}, and consider spanning trees in such graphs. Using the notion of total excess, we prove that if G is 1−ε n−2+ε -tough for an integer n ≥ 2 and a real number ε with 2 |V (G)| ≤ ε ≤ 1, then G has a spanning tree T such that ∑ v∈V (G) max{0, degT (v)− n} ≤ ε|V (G)| − 2. We 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 2 |V (G)| ≤ ε ≤ 1, which satisfy t ≥ 1−ε n−2+ε .