The Disjoint \(1\)-Factors of \((d, d+1)\)-Graphs

A graph \(G\) is called \((d,d+1)\)-graph if the degree of every vertex of \(G\) is either \(d\) or \(d+1\). In this paper, the following results are proved:
A \((d,d+1)\)-graph \(G\) of order \(2n\) with no \(1\)-factor and no odd component, satisfies \(|V(G)| \geq 3d+4\);
A \((d,d+1)\)-graph \(G\) of order \(2n\) with \(d(G) \geq n\), contains at least \([(n+2)/{3}] + (d-n)\) edge disjoint \(1\)-factors.
These results generalize the theorems due to W. D. Wallis, A. I. W. Hilton and C. Q. Zhang.