Isolated toughness and path-factor uniform graphs

Abstract

A P≥k-factor of a graph G is a spanning subgraph of G whose components are paths of order at least k. We say that a graph G is P≥k-factor covered if for every edge e ∈ E(G), G admits a P≥k-factor that contains e; and we say that a graph G is P≥k-factor uniform if for every edge e ∈ E(G), the graph G−e is P≥k-factor covered. In other words, G is P≥k-factor uniform if for every pair of edges e1, e2 ∈ E(G), G admits a P≥k-factor that contains e1 and avoids e2. In this article, we testify that (1) a 3-edge-connected graph G is P≥k-factor uniform if its isolated toughness I(G) > 1; (2) a 3-edge-connected graph G is P≥k-factor uniform if its isolated toughness I(G) > 2. Furthermore, we explain that these conditions on isolated toughness and edge-connectivity in our main results are best possible in some sense.