Degree sum conditions for path-factor uniform graphs

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->= k-factor of G if its each component admits at least k vertices, where k >= 2. A graph G is called a P->= k-factor uniform graph if for any two different edges e(1) and e(2) of G, G admits a P->= k-factor containing e(1) and avoiding e(2). The degree sum of G is defined by sigma(k)(G) = min(X subset of V(G)) {Sigma(x epsilon X) dG(x) : X is an independent set of k vertices}. In this paper, we give two degree sum conditions for a graph to be a P->= 2-factor uniform graph and a P->= 3-factor uniform graph, respectively.