\(k\)-factors in regular graphs and edge-connectivity (Q950664)

Let \(\ell, r,m,n\) be integers such that \(r\geq 4\), \(2 \leq \ell \leq r\), \(0 \leq m, n\) and \(G\) be an \(r\)-regular \(\ell\)-edge-connected graph. \(G\) is an \((m,n;k)\)-factor graph if \(m \leq k \leq \frac{r}{2}\) and for each disjoint pair \(E_1,E_2 \subseteq E(G)\) with \(|E_1|=m\), \(|E_2|=n\), \(G\) has a \(k\)-factor \(F\) such that \(E_1 \subseteq E(F)\) and \(E_2 \cap E(F) = \emptyset\). It is shown that each of the following conditions: {\parindent=8mm \begin{itemize}\item[(i)]\(m=0\) and \(n\leq\lceil\frac{\ell+r}{2}\rceil -k\), \item[(ii)]\(1\leq m\leq\frac{k}{2}\) and \(m+n\leq\lceil\frac{\ell}{2}\rceil\), \item[(iii)]\(\frac{k}{2}< m\leq{k}\) and \(m+n\leq k+\lceil\frac{\ell-r}{2}\rceil\) \end{itemize}} is sufficient for an \(r\)-regular \(\ell\)-edge-connected graph to be either an \((m,n;k)\)-factor or nearly bipartite. This generalizes previous similar result by \textit{Y. Egawa} and \textit{K. Kotani} [``\(k\)-factors containing and avoiding specified sets of edges'', AKCE Int. J. Graphs Comb. 4, No. 1, 11--19 (2007; Zbl 1142.05066)] on \(r\)-regular \(r\)-edge-connected graphs.