Sufficient conditions for component factors in a graph

Abstract

Let G be a graph and \(\mathcal {H}\) be a set of connected graphs. A spanning subgraph H of G is called an \(\mathcal {H}\)–factor if each component of H is isomorphic to a member of \(\mathcal {H}\). In this paper, we first present a lower bound on the size (resp. the spectral radius) of G to guarantee that G has a \(\{P_2,\, C_n: n\ge 3\}\)–factor (or a perfect k–matching for even k) and construct extremal graphs to show all this bounds are best possible. We then provide a lower bound on the signless laplacian spectral radius of G to ensure that G has a \(\{K_{1,j}:1\le j\le k\}\)–factor, where \(k\ge 2 \) is an integer. Moreover, we also provide some Laplacian eigenvalue (resp. toughness) conditions for the existence of \(\{P_2,\, C_{n}:n\ge 3\}\)–factor, \(P_{\ge 3}\)–factor and \(\{K_{1,j}: 1\le j\le k\}\)–factor in G, respectively. Some of our results extend or improve the related existing results.