Remarks on component factors in graphs

Abstract

For a family of connected graphs ℱ, a spanning subgraph H of a graph G is called an ℱ-factor of G if its each component is isomorphic to an element of ℱ. In particular, H is called an Sk-factor of G if ℱ = {K1,1, K1,2,…,K1,k}, where integer k ≥ 2; H is called a P≥3-factor of G if every component in ℱ is a path of order at least three. As an extension of Sk-factors, the induced star-factor (i.e., ISk-factor) is a spanning subgraph each component of which is an induced subgraph isomorphic to some graph in ℱ = {K1,1, K1,2,…,K1,k}. In this paper, we firstly prove that a graph G has an Sk-factor if and only if its isolated toughness I(G) ≥ 1/k. Secondly, we prove that a planar graphs G has an S2-factors if its minimum degree δ(G) ≥ 3. Thirdly, we give two sufficient conditions for graphs with ISk-factors by toughness and minimum degree, respectively. Additionally, we obtain three special classes of graphs admitting P≥3-factors.