Binding number and degree conditions for path-factors in graphs

Abstract

For a graph $G$ and an integer $k\geq 2$, a $P_{\geq k}$-factor of $G$ is a spanning subgraph of $G$ with each component isomorphic to some path of order at least $k$.
A graph $G$ is $P_{\geq k}$-factor uniform if for any distinct edges $e_1$ and $e_2$, $G$ admits a $P_{\geq k}$-factor including $e_1$ and excluding $e_2$. For a non-negative integer $n$, $G$ is $(P_{\geq k},n)$-critical uniform if for any $V'\subseteq V(G)$ with $|V'|=n$, $G-V'$ is $P_{\geq k}$-factor uniform. $G$ is $(P_{\geq k},n)$-critical deleted if for any $V'\subseteq V(G)$ with $|V'|=n$ and $e\in E(G-V')$, $G-V'-e$ contains a $P_{\geq k}$-factor. In this note, we give some binding number conditions and degree conditions for a graph to be $(P_{\geq 2},n)$-critical uniform and $(P_{\geq 3},n)$-critical deleted, which improve some known results.