MAXIMUM PACKINGS AND MINIMUM COVERINGS OF MULTIGRAPHS WITH PATHS AND STARS

Abstract

Let $F$, $G$, and $H$ be multigraphs. An $(F,G)$-decomposition of $H$ is an edge decomposition of $H$ into copies of $F$ and $G$ using at least one of each. For subgraphs $L$ and $R$ of $H$, an $(F,G)$-packing of $H$ with leave $L$ is an $(F,G)$-decomposition of $H-E(L)$, and an $(F,G)$-covering of $H$ with padding $R$ is an $(F,G)$-decomposition of $H+E(R)$. A maximum $(F,G)$-packing of $H$ is an $(F,G)$-packing of $H$ with a minimum leave. A minimum $(F,G)$-covering of $H$ is an $(F,G)$-covering of $H$ with a minimum padding. Let $k$ be a positive integer. A $k$-path, denoted by $P_k$, is a path on $k$ vertices. A $k$-star, denoted by $S_k$, is a star with $k$ edges. In this paper, we obtain a maximum $(P_{k+1},S_{k})$-packing of $\lambda K_n$, which has a leave of size $\lt k$, and a minimum $(P_{k+1},S_{k})$-covering of $\lambda K_n$, which has a padding of size $\lt k$. A similar result for $\lambda K_{n,n}$ is also obtained. As corollaries, necessary and sufficient conditions for the existence of $(P_{k+1},S_k)$-decompositions of both $\lambda K_n$ and $\lambda K_{n,n}$ are given.