Some results on Steiner decomposition number of graphs

Abstract

Let $G$ be a connected graph with Steiner number $s(G)$. A decomposition $\pi=\{G_1, G_2,..., G_n\}$ is said to be a Steiner decomposition if $s(G_i)=s(G)$ for all $i\:(1\leq i\leq n)$. The maximum cardinality obtained for the Steiner decomposition $\pi$ of $G$ is called the Steiner decomposition number of $G$ and is denoted by $\pi_{st}(G)$. In this paper we present a relation between Steiner decomposition number and independence number of $G.$ Steiner decomposition number for some power of paths are discussed. It is also shown that given any pair $m,n$ of positive integers with $m\geq2$ there exists a connected graph $G$ such that $s(G)=m$ and $\pi_{st}(G)=n.$