Degree conditions for path-factor critical deleted or covered graphs

Abstract

A path-factor of a graph G is a spanning subgraph of G whose components are paths. A P≥d-factor of a graph G is a path-factor of G whose components are paths with at least d vertices, where d is an integer with d ≥ 2. A graph G is P≥d-factor covered if for any e ∈ E(G), G admits a P≥d-factor including e. A graph G is (P≥d, n)-factor critical deleted if for any Q ⊆ V(G) with |Q| = n and any e ∈ E(G − Q), G – Q − e has a P≥d-factor. A graph G is (P≥d, n)-factor critical covered if for any Q ⊆ V(G) with |Q| = n, G − Q is a P≥d-factor covered graph. In this paper, we verify that (1) an (n + t + 2)-connected graph G of order p with p ≥ 4t + n + 7 is (P≥3, n)-factor critical deleted if max $ \left\{{d}_G({v}_1),{d}_G({v}_2),\dots,{d}_G({v}_{2t+1})\right\}\ge \frac{p+2n}{3}$ for any independent set {v1, v2, …, v2t+1} of G, where n and t are two nonnegative integers with t ≥ 1; (2) an (n + t + 1)-connected graph G of order p with p ≥ 4t + n + 5 is (P≥3, n)-factor critical covered if max $ \left\{{d}_G({v}_1),{d}_G({v}_2),\dots,{d}_G({v}_{2t+1})\right\}\ge \frac{p+2n+2}{3}$ for any independent set {v1, v2, …, v2t+1} of G, where n and t are two nonnegative integers with t ≥ 1.