TOUGHNESS OF GRAPHS AND K-FACTORS WITH GIVEN PROPERTIES

Chvatal [1] presented the conjecture that every k-tough graph G has a k-factor if G satisfies trivial necessary conditions. The truth of Chvatal's conjecture was proved by Enomoto et al. [2]. Here we prove the following stronger results: every k-tough graph satisfying trivial necesary conditions has a k-factor which contains an arbitrarily given edge if k greater-than-or-equal-to 2, and also has a k-factor which does not contain an arbitrarily given edge (k greater-than-or-equal-to 1).