Component factors and degree sum conditions in graphs

For a set A of connected graphs, an A -factor is a spanning subgraph of a graph, whose connected components are isomorphic to graphs from the set A. An A -factor is also referred as a component factor. A graph G is called an (A, m) -factor deleted graph if for every E0 ⊆ E (G) with |E0| = m, G−E0 admits an A -factor. A graph G is called an (A, l) -factor critical graph if for every V0 ⊆V (G) with |V0| = l, G−V0 admits an A -factor. Let m, l and k be three positive integers with k ≥ 2, and write F = P2, C3, P5, T (3) and H = K1, 1, K1, 2,. . . , K1, k, T (2k+1), where T (3) and T (2k+1) are two special families of trees. Inspired by finding a sufficient condition to check for the existence of path-factors with some special restraints, we focus on the sufficient conditions based on a graphic parameter called degree sum: σk (G) =minₗ ₕ (₆) [EQUATION]dG (x): X is an independent set of k vertices. In this article, we verify that (i) an (l+2) -connected graph G of order n is an (F, l) -factor critical graph if σ3 (G) 6n+9l5; (ii) a (2m+1) -connected graph G of order n is an (F, m) -factor deleted graph if σm+2 (G) 65n; (iii) an (l+2) -connected graph G of order n is an (H, l) -factor critical graph if σ2k+1 (G) 6n+ (6k+3) l2k+3; (iv) a (2m+1) -connected graph G of order n is an (H, m) -factor deleted graph if σm+2 (G) 6n2k+3.