Degree sum conditions for path-factor uniform graphs

Abstract

A spanning subgraph of a graph G is called a path-factor of G if its each component is a path. A path-factor is called a $P_{\ge k}$-factor of G if its each component admits at least k vertices, where $k\ge 2$. A graph G is called a $P_{\ge k}$-factor uniform graph if for any two different edges $e_1$ and $e_2$ of G, G admits a $P_{\ge k}$-factor containing $e_1$ and avoiding $e_2$. The degree sum of G is defined by

$$\begin{aligned} \sigma _k(G)=\min _{X\subseteq V(G)}\Big \{\sum _{x\in X}d_G(x): X~\mathrm {is~an~independent~set~of}~k~\textrm{vertices}\Big \}. \end{aligned}$$

In this paper, we give two degree sum conditions for a graph to be a $P_{\ge 2}$-factor uniform graph and a $P_{\ge 3}$-factor uniform graph, respectively.