Some sufficient conditions for path-factor uniform graphs

Abstract

For a set H of connected graphs, a spanning subgraph H of G is called an H-factor of G if each component of H is isomorphic to an element of H. A graph G is called an H-factor uniform graph if for any two edges e1 and e2 of G, G has an H-factor covering e1 and excluding e2. Let each component in H be a path with at least d vertices, where d ≥ 2 is an integer. Then an H-factor and an H-factor uniform graph are called a P ≥d -factor and a P ≥d -factor uniform graph, respectively. In this article, we verify that (i) a 2-edge-connected graph G is a P ≥3 -factor uniform graph if δ(G) > α(G)+4 2 ; (ii) a (k + 2)-connected graph G of order n with n ≥ 5k + 3 − 3 5γ−1 is a P ≥3 -factor uniform graph if |NG(A)| > γ(n − 3k − 2) + k + 2 for any independent set A of G with |A| = ⌊γ(2k + 1)⌋, where k is a positive integer and γ is a real number with 1 3 ≤ γ ≤ 1.