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 $$\frac{1}{3}\le \gamma \le 1$$ .