Krenn, Gu and Zeilinger initiated the study of PMValid edge-colourings because of its connection to a problem from quantum physics. A graph is defined to have a PMValid $k$-edge-colouring if it admits a $k$-edge-colouring (i.e. an edge colouring with $k$-colours) with the property that all perfect matchings are monochromatic and each of the $k$ colour classes contain at least one perfect matching. The matching index of a graph $G$, $\mu(G)$ is defined as the maximum value of $k$ for which $G$ admits a PMValid $k$-edge-colouring. It is easy to see that $\mu(G)\geq 1$ if and only if $G$ has a perfect matching (due to the trivial $1$-edge-colouring which is PMValid). Bogdanov observed that for all graphs non-isomorphic to $K_4$, $\mu(G)\leq 2$ and $\mu(K_4)=3$. However, the characterisation of graphs for which $\mu(G)=1$ and $\mu(G)=2$ is not known. In this work, we answer this question. Using this characterisation, we also give a fast algorithm to compute $\mu(G)$ of a graph $G$. In view of our work, the structure of PMValid $k$-edge-colourable graphs is now fully understood for all $k$. Our characterisation, also has an implication to the aforementioned quantum physics problem. In particular, it settles a conjecture of Krenn and Gu for a sub-class of graphs.

翻译：Krenn、Gu和Zeilinger因与量子物理问题的关联，率先开展了PMValid边染色研究。定义图具有PMValid $k$-边染色，是指其存在一种$k$-边染色（即使用$k$种颜色的边染色），满足所有完美匹配均为单色且每个颜色类至少包含一个完美匹配。图$G$的匹配指数$\mu(G)$定义为使$G$具有PMValid $k$-边染色的最大$k$值。易见，$\mu(G)\geq 1$当且仅当$G$存在完美匹配（由平凡的PMValid $1$-边染色可得）。Bogdanov发现，对于所有非同构于$K_4$的图，$\mu(G)\leq 2$，而$\mu(K_4)=3$。然而，满足$\mu(G)=1$和$\mu(G)=2$的图的特征刻画此前未知。本文中，我们回答了这一问题。借助该刻画，我们还给出了计算图$G$的$\mu(G)$值的快速算法。基于本文工作，对所有$k$值，具有PMValid $k$-边染色的图的结构现已完全理解。该刻画还对前述量子物理问题具有启示意义，特别地，它解决了一个关于图子类的Krenn-Gu猜想。