Egawa, Yoshimi; Enomoto, Hikoe; Tokushige, Norihide Graph decompositions through prescribed vertices without isolates. (English) Zbl 1072.05558 Ars Comb. 62, 189-205 (2002). Summary: 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 \(v_1,\dots ,v_k\) 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 \(i\), \(1 \leq i \leq k\). Cited in 1 Document MSC: 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) Keywords:graph decomposition; isolated vertex; minimum degree PDFBibTeX XMLCite \textit{Y. Egawa} et al., Ars Comb. 62, 189--205 (2002; Zbl 1072.05558)