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 E ' subset of E(G) with |E '| = m, G - E ' admits an A-factor. A graph G is called an (A, l)-factor critical graph if for every V ' subset of V (G) with |V '| = l, G - V ' admits an A-factor. Let m, l and k be three positive integers with k >= 2, and write F = {P2, C3, P5, J (3)} and H = {K1,1, K1,2, . . . , K1,k, J (2k + 1)}, where J (3) and J (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: sigma(k)(G) = min(X subset of V(G)) {Sigma(x is an element of X)d(G)(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 sigma(3)(G) >= 6n+9l/5 ; (ii) a (2m + 1)-connected graph G of order n is an (F, m)-factor deleted graph if sigma(m+2)(G) >= 6/5n ; (iii) an (l + 2)-connected graph G of order n is an (H, l)-factor critical graph if sigma(2k+1)(G) >= 6n+(6k+3)l/2k+3 ; (iv) a (2m + 1)-connected graph G of order n is an (H, m)-factor deleted graph if sigma(m+2)(G) >= 6n/2k+3 .