arXiv subjects
Publications and source records attributed to Dingjun Lou.
We determine the minimum size of $n$-factor-critical graphs and that of $k$-extendable bipartite graphs, by considering Harary graphs and related graphs. Moreover, we determine the minimum size of $k$-extendable non-bipartite graphs for $k=1,\ 2$, and pose a related conjecture for general $k$.
We study $M$-alternating Hamilton paths and $M$-alternating Hamilton cycles in a simple connected graph $G$ on $\nu$ vertices with a perfect matching $M$. Let $G$ be a bipartite graph, we prove that if for any two vertices $x$ and $y$ in different parts of $G$, $d(x)+d(y)\geq \nu/2+2$, then $G$ has an $M$-alternating Hamilton cycle. For general graphs, a condition for the existence of an $M$-alternating Hamilton path starting and ending with edges in $M$ is put forward. Then we prove that if $\kappa(G)\geq\nu/2$, where $\kappa(G)$ denotes the connectivity of $G$, then $G$ has an $M$-alternating Hamilton cycle or belongs to one class of exceptional graphs. Lou and Yu \cite{LY} have proved that every $k$-extendable graph $H$ with $k\geq\nu/4$ is bipartite or satisfies $\kappa(H)\geq 2k$. Combining this result with those we obtain we prove the existence of $M$-alternating Hamilton cycles in $H$.
In this paper, we introduce a corresponding between bipartite graphs with a perfect matching and digraphs, which implicates an equivalent relation between the extendibility of bipartite graphs and the strongly connectivity of digraphs. Such an equivalent relation explains the similar results on $k$-extendable bipartite graphs and $k$-strong digraphs. We also study the relation among $k$-extendable bipartite graphs, $k$-strong digraphs and combinatorial matrices. For bipartite graphs that are not 1-extendable and digraphs that are not strong, we prove that the elementary components and strong components are counterparts.
In this paper, we show that if $k\geq (\nu+2)/4$, where $\nu$ denotes the order of a graph, a non-bipartite graph $G$ is $k$-extendable if and only if it is $2k$-factor-critical. If $k\geq (\nu-3)/4$, a graph $G$ is $k\ 1/2$-extendable if and only if it is $(2k+1)$-factor-critical. We also give examples to show that the two bounds are best possible. Our results are answers to a problem posted by Favaron [3] and Yu [11].
In this paper, we give a sufficient and necessary condition for a $k$-extendable graph to be $2k$-factor-critical when $k=\nu/4$, and prove some results on independence numbers in $n$-factor-critical graphs and $k\frac{1}{2}$-extendable graphs.