Packing 2- and 3-stars into (2,3)-regular graphs

Abstract

Let $i$ be a positive integer, an $i$-star denoted by $S_i$ is a complete bipartite graph $K_{1,i}$. An $\{S_2,S_3\}$-packing of a graph $G$ is a collection of vertex-disjoint subgraphs of $G$ in which each subgraph is a 2-star or a 3-star. The maximum $\{S_2,S_3\}$-packing problem is to find an $\{S_2,S_3\}$-packing of a given graph containing the maximum number of vertices. The $\{S_2,S_3\}$-factor problem is to answer whether there is an $\{S_2,S_3\}$-packing containing all vertices of the given graph. The $\{S_2,S_3\}$-factor problem is NP-complete in cubic graphs. In this paper we design a quadratic-time algorithm for finding an $\{S_2,S_3\}$-packing of $G$ that covers at least thirteen-sixteenths of its vertices with only a few exceptions. We also present some $(2,3)$-regular graphs with their maximum $\{S_2,S_3\}$-packings covering exactly thirteen-sixteenths of their vertices.