Sufficient conditions for the existence of path-factors with given properties

Abstract

A spanning subgraph $H$ of a graph $G$ is called a $P_{\ge k}$-factor of $G$ if every component of $H$ is isomorphic to a path of order at least $k$, where $k\ge 2$ is an integer. A graph $G$ is called a $(P_{\ge k},l)$-factor critical graph if $G-V^{\prime }$ contains a $P_{\ge k}$-factor for any $V^{\prime }\subseteq V(G)$ with $|V^{\prime }|=l$. A graph $G$ is called a $(P_{\ge k},m)$-factor deleted graph if $G-E^{\prime }$ has a $P_{\ge k}$-factor for any $E^{\prime }\subseteq E(G)$ with $|E^{\prime }|=m$. Intuitively, if a graph is dense enough, it will have a $P_{\ge 3}$-factor. In this paper, we give some sufficient conditions for a graph to be a $(P_{\ge 3},l)$-factor critical graph or a $(P_{\ge 3},m)$-factor deleted graph. In this paper, we demonstrate that (i) $G$ is a $(P_{\ge 3},l)$-factor critical graph if its sun toughness $s(G)>\frac{l+1}{3}$ and $\kappa (G)\ge l+2$. (ii) $G$ is a $(P_{\ge 3},l)$-factor critical graph if its degree sum ${\sigma }_3(G)\ge n+2l$ and $\kappa (G)\ge l+1$. (iii) $G$ is a $(P_{\ge 3},m)$-factor deleted graph if its sun toughness $s(G)\ge \frac{m+1}{m+2}$ and $\kappa (G)\ge 2m+1$. (iv) $G$ is a $(P_{\ge 3},m)$-factor deleted graph if its degree sum ${\sigma }_3(G)\ge n+2m$ and $\kappa (G)\ge 2m+1$.