Degree sum and restricted {P2,P5}-factor in graphs

Abstract

"For a graph $G$, a spanning subgraph $F$ of $G$ is called a $\{P_2,P_5\}$-factor if every component of $F$ is isomorphic to $P_2$ or $P_5$, where $P_i$ denotes the path of order $i$. A graph $G$ is called a $(\{P_2,P_5\},k)$-factor critical graph if $G-V'$ contains a $\{P_2,P_5\}$-factor for any $V'\subseteq V(G)$ with $|V'|=k$. A graph $G$ is called a $(\{P_2,P_5\},m)$-factor deleted graph if $G-E'$ has a $\{P_2,P_5\}$-factor for any $E'\subseteq E(G)$ with $|E'|=m$. The degree sum of $G$ is defined by $$\sigma_{r+1}(G)=\min_{X\subseteq V(G)}\Big\{\sum_{x\in X}d_G(x): X~\mathrm{is~an~independent~set~of}~r+1~\mathrm{vertices}\Big\}.$$ In this paper, using degree sum conditions, we demonstrate that (i) $G$ is a $(\{P_2,P_5\},k)$-factor critical graph if $\sigma_{r+1}(G)>\frac{(3n+4k-2)(r+1)}{7}$ and $\kappa(G)\geq k+r$; (ii) $G$ is a $(\{P_2,P_5\},m)$-factor deleted graph if $\sigma_{r+1}(G)>\frac{(3n+2m-2)(r+1)}{7}$ and $\kappa(G)\geq\frac{5m}{4}+r$."