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arXiv · 2211.03133

Minimizing the number of matchings of fixed size in a $K_s$-saturated graph

Jiejing Feng·Doudou Hei·Xinmin Hou
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Abstract

For a fixed graph $F$, a graph $G$ is said to be $F$-saturated if $G$ does not contain a subgraph isomorphic to $F$ but does contain $F$ after the addition of any new edge. Let $M_k$ be a matching consisting of $k$ edges and $S_{n,k}$ be the join graph of a complete graph $K_k$ and an empty graph $\overline{K_{n-k}}$. In this paper, we prove that for $s \geq3$ and $k\geq 2$, $S_{n,s-2}$ contains the minimum number of $M_k$ among all $n$-vertex $K_s$-saturated graphs for sufficiently large $n$, and when $k \leq s-2$, it is the unique extremal graph. In addition, we also show that $S_{n,1}$ is the unique extremal graph when $k=2$ and $s=3$.

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Jiejing Feng, Doudou Hei, Xinmin Hou. 2022-11-06. Minimizing the number of matchings of fixed size in a $K_s$-saturated graph. https://arxiv.org/abs/2211.03133

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