Graph Decompositions Through Prescribed Vertices Without Isolates

Abstract

Let $G$ be a graph of order $n$, and let $n=\sum _{i=1}^k{a}^i$ be a partition of $n$ with $a_i\ge 2$. Let $v_1,\dots ,v_k$ be given distinct vertices of $G$. Suppose that the minimum degree of $G$ is at least $3k$. In this paper, we prove that there exists a decomposition of the vertex set $V(G)=\bigcup _{i=1}^k{A}_i$ such that $|A_i|=a_i$, $v_i\in A_i$, and the subgraph induced by $A_i$ contains no isolated vertices for all $i,1\le i\le k$.