报告人：鲁红亮博士，西安交通大学

报告时间：2015年12月9日（周三）15点

报告地点：离散数学研究中心二层报告厅

报告题目：Disjoint perfect matchings in semi-regular graphs

报告摘要：Let $n\\ge 34$ be an even integer, and $D_n=2\\lceil n/4 \ceil-1$. In this paper, we prove that every $\\{D_n,\\,D_n+1\\}$-regular graph of order $n$ contains $\\lceil n/4 \ceil$ disjoint perfect matchings. This result is sharp in the sense that (i) there exists a $\\{D_n,\\,D_n+1\\}$-regular graph containing exactly $\\lceil n/4 \ceil$ disjoint perfect matchings, and that (ii) there exists a $\\{D_n-1,\\,D_n\\}$-regular graph without perfect matchings for each $n$. As a consequence, for any integer $D\\ge D_n$, every $\\{D,\\,D+1\\}$-regular graph of order $n$ contains $\\lceil (D+1)/2 \ceil$ disjoint perfect matchings. This extends Csaba et~al.'s breathtaking result that every $D$-regular graph of sufficiently large order is $1$-factorizable, generalizes Zhang and Zhu's result that every $D_n$-regular graph of order $n$ contains $\\lceil n/4 \ceil$ disjoint perfect matchings, and improves Hou's result that for all $k\\ge n/2$, every $\\{k,\\,k+1\\}$-regular graph of order $n$ contains $(\\lfloor n/3\floor+1+k-n/2)$ disjoint perfect matchings.

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