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arXiv · 1909.09093

New results relating independence and matchings

Yair Caro·Randy Davila·Ryan Pepper
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Abstract

In this paper we study relationships between the \emph{matching number}, written $\mu(G)$, and the \emph{independence number}, written $\alpha(G)$. Our first main result is to show \[ \alpha(G) \le \mu(G) + |X| - \mu(G[N_G[X]]), \] where $X$ is \emph{any} intersection of maximum independent sets in $G$. Our second main result is to show \[ \delta(G)\alpha(G) \le \Delta(G)\mu(G), \] where $\delta(G)$ and $\Delta(G)$ denote the minimum and maximum vertex degrees of $G$, respectively. These results improve on and generalize known relations between $\mu(G)$ and $\alpha(G)$. Further, we also give examples showing these improvements.

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Yair Caro, Randy Davila, Ryan Pepper. 2019-09-19. New results relating independence and matchings. https://arxiv.org/abs/1909.09093

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