论文标题
分裂匹配和多种群体的Ryser-Brualdi-Stein猜想
Splitting matchings and the Ryser-Brualdi-Stein conjecture for multisets
论文作者
论文摘要
我们研究了多式图像，其边缘设备是三个完美匹配的结合，$ m_1 $，$ m_2 $和$ m_3 $。给定这样的图形$ g $和任何$ a_1，a_2，a_3 \ in \ mathbb {n} $，带有$ a_1+a_2+a_2+a_2+a_3 \ leq n-2 $，我们显示出存在$ g $的$ m $ g $，$ | m \ cap m_i | cap m_i | = a_i | = a_i $ |总体而言，最大的可能是定理中的$ n-2 $。但是，我们推测，如果$ g $是双方的，那么$ N-2 $被$ n-1 $取代的结果相同。我们提供的结构表明这种结果会很紧。我们还猜想了具有颜色多重性的Ryser-Brualdi-Stein猜想。
We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3\in \mathbb{N}$ with $a_1+a_2+a_3\leq n-2$, we show there exists a matching $M$ of $G$ with $|M\cap M_i|=a_i$ for each $i\in \{1,2,3\}$. The bound $n-2$ in the theorem is best possible in general. We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.
