Some sufficient conditions for path-factor uniform graphs.

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 e 1 and e 2 of G, G has an H -factor covering e 1 and excluding e 2 . 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 ≥ 5 k + 3 - 3 5 γ - 1 is a P ≥ 3 -factor uniform graph if | N G (A) | > γ (n - 3 k - 2) + k + 2 for any independent set A of G with | A | = ⌊ γ (2 k + 1) ⌋ , where k is a positive integer and γ is a real number with 1 3 ≤ γ ≤ 1 . [ABSTRACT FROM AUTHOR]