Toughness of Graphs and \([a, b]\)-Factors with Prescribed Properties

Chvátal conjectured that if \(G\) is a \(k\)-tough graph and \(k|V(G)|\) is even, then \(G\) has a \(k\)-factor. In \([5\) it was proved that Chvátal’s conjecture is true. Katerinis\([2]\) presented a toughness condition for a graph to have an \([a, b]\)-factor. In this paper, we prove a stronger result: every \((a – 1 + a/b)\)-tough graph satisfying all necessary conditions has an \([a, b]\)-factor containing any given edge and another \([a, b]\)-factor excluding it. We also discuss some special cases of the above result.