Some sufficient conditions for path-factor uniform graphs

Abstract

For a set $\mathcal {H}$ of connected graphs, a spanning subgraph H of G is called an $\mathcal {H}$-factor of G if each component of H is isomorphic to an element of $\mathcal {H}$. A graph G is called an $\mathcal {H}$-factor uniform graph if for any two edges $e_1$ and $e_2$ of G, G has an $\mathcal {H}$-factor covering $e_1$ and excluding $e_2$. Let each component in $\mathcal {H}$ be a path with at least d vertices, where $d\ge 2$ is an integer. Then an $\mathcal {H}$-factor and an $\mathcal {H}$-factor uniform graph are called a $P_{\ge d}$-factor and a $P_{\ge d}$-factor uniform graph, respectively. In this article, we verify that (i) a 2-edge-connected graph G is a $P_{\ge 3}$-factor uniform graph if $\delta (G)>\frac{\alpha (G)+4}{2}$; (ii) a $(k+2)$-connected graph G of order n with $n\ge 5k+3-\frac{3}{5\gamma -1}$ is a $P_{\ge 3}$-factor uniform graph if $|N_G(A)|>\gamma (n-3k-2)+k+2$ for any independent set A of G with $|A|=\lfloor \gamma (2k+1)\rfloor$, where k is a positive integer and $\gamma$ is a real number with $\frac{1}{3}\le \gamma \le 1$.