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Connected detachments of graphs and generalized Euler trails. (English) Zbl 0574.05042
A graph F is called a detachment of a given graph G, if $$E(F)=E(G)$$, and there exists a function p from V(F) onto V(G) such that, for each edge $$e\in E(G)$$, the vertices joined by e in G are the images under p of the vertices joined by e in F. Roughly speaking, the graph F is obtained from G by splitting some or all of its vertices into more than one vertex. The paper is a continuation of the author’s earlier paper ”Acyclic Detachments of Graphs” in Graph theory and combinatorics, Proc. Conf., The Open Univ./Engl. 1978, Res. Notes Math. 34, 87-97 (1979; Zbl 0464.05042). In the paper, the conditions are given such that graph G has a connected detachment in which each vertex of G splits into a prescribed number of vertices with prescribed valencies. Similar conditions are also given for the k-connected case. Using these conditions, two corollaries, which generalize the well-known closed Euler trail theorem in two different ways, are presented.
Reviewer: Feng Tian

##### MSC:
 05C99 Graph theory 05C38 Paths and cycles
##### Keywords:
connected detachment; closed Euler trail theorem
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