Graph decompositions without isolates
A. Frank conjectured that if \(G=(V,E)\) is a connected graph with minimal degree \(\geq k\) and \(a_ 1,...,a_ k\geq 2\) are integers with \(\sum a_ i=| V|\), then V may be decomposed into \(A_ 1,...,A_ k\), such that \(| A_ i| =a_ i\) and the subgraph induced by \(A_ i\) in G has no isolated vertices \((i=1,...,k)\). The case \(k=2\) has been proved by \textit{S. B. Maurer} [J. Comb. Theory, Ser. B 27, 294-319 (1979; Zbl 0413.05009)]. In this article the author proves the conjecture for \(k=3\) and for the case \(a_ 1=...=a_{k-1}=2\). Additionally he proves a related characterization of k-connected graphs which was independently proved by \textit{E. Györi} [Combinatorica 1, 263-273 (1981; Zbl 0494.05041)].
scientific article; zbMATH DE number 3849280