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 \(a_i \geq 2\). Let \(v_1, \ldots, v_k\) 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 \(i, 1 \leq i \leq k\).