arXiv 1507.062542015-07math.COcs.DM

距离正则图中匹配的最大割与可扩性

Max-cut and extendability of matchings in distance-regular graphs

Sebastian M. Cioabă, Jack H. Koolen, Weiqiang Li

备注：: 18 pages, accepted to European Journal of Combinatorics
DOI：: 10.1016/j.ejc.2017.01.001

摘要：

设$G$是阶为$v$、大小为$e$的距离正则图.本文证明了$G$的最大割至多为$e(1-1/g)$，其中$g$是$G$的奇数围长.这个结果意味着$G$的独立数至多为$\frac{v}{2}(1-1/g)$。我们利用这一事实也研究了距离正则图中匹配的可拓性。一个偶数阶图$G$称为$t$ -可扩的，如果它包含一个完美匹配，$t<v/2$和任何$t$边的匹配都包含在某个完美匹配中. $G$的可扩展性是使得$G$是$t$可扩展的最大$t$。推广了强正则图的已有结果，证明了所有直径为$D\geq 3$的距离正则图都是$2$ -可扩的.我们还得到了各种下界的距离正则图的价$k$，$λ$和$μ$，$λ$是任何两个相邻顶点的公共邻居的数量和$μ$是任何两个顶点的公共邻居的距离2依赖于可扩展性。

英文摘要：

Let $G$ be a distance-regular graph of order $v$ and size $e$. In this paper, we show that the max-cut in $G$ is at most $e(1-1/g)$, where $g$ is the odd girth of $G$. This result implies that the independence number of $G$ is at most $\frac{v}{2}(1-1/g)$. We use this fact to also study the extendability of matchings in distance-regular graphs. A graph $G$ of even order $v$ is called $t$-extendable if it contains a perfect matching, $t<v/2$ and any matching of $t$ edges is contained in some perfect matching. The extendability of $G$ is the maximum $t$ such that $G$ is $t$-extendable. We generalize previous results on strongly regular graphs and show that all distance-regular graphs with diameter $D\geq 3$ are $2$-extendable. We also obtain various lower bounds for the extendability of distance-regular graphs of valency $k$ that depend on $k$, $λ$ and $μ$, where $λ$ is the number of common neighbors of any two adjacent vertices and $μ$ is the number of common neighbors of any two vertices in distance two.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文