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 \(k \geq 2\), and also has a \(k\)-factor which does not contain an arbitrarily given edge \(v(k \geq 1)\).