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arXiv · 2005.00671

Pairs of disjoint matchings and related classes of graphs

Huizheng·Guo·Kieran Kaempen·Zhengda Mo·Sam Qunell·Joe Rogge·Chao Song·Anush Tserunyan·Jenna Zomback
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Abstract

For a finite graph $G$, we study the maximum $2$-edge colorable subgraph problem and a related ratio $\frac{\mu(G)}{\nu(G)}$, where $\nu(G)$ is the matching number of $G$, and $\mu(G)$ is the size of the largest matching in any pair $(H,H')$ of disjoint matchings maximizing $|H| + |H'|$ (equivalently, forming a maximum $2$-edge colorable subgraph). Previously, it was shown that $\frac{4}{5} \le \frac{\mu(G)}{\nu(G)} \le 1$, and the class of graphs achieving $\frac{4}{5}$ was completely characterized. We show here that any rational number between $\frac{4}{5}$ and $1$ can be achieved by a connected graph. Furthermore, we prove that every graph with ratio less than $1$ must admit special subgraphs.

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Huizheng, Guo, Kieran Kaempen, Zhengda Mo, Sam Qunell, Joe Rogge, Chao Song, Anush Tserunyan, Jenna Zomback. 2020-05-02. Pairs of disjoint matchings and related classes of graphs. https://doi.org/10.2140/involve.2023.16.249

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