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Amalgamations of almost regular edge-colourings of simple graphs. (English) Zbl 0654.05031
Summary: A finite graph F is a detachment of a finite graph G if G can be obtained from F by partitioning V(F) into disjoint sets $$S_ 1,...,S_ n$$ and identifying the vertices in $$S_ i$$ to form a single vertex $$\alpha_ i$$ for $$i=1,...,n$$. Thus $$E(F)=E(G)$$ and an edge which joins an element of $$S_ i$$ to an element of $$S_ j$$ in F will join $$\alpha_ i$$ to $$\alpha_ j$$ in G. If L is a subset of E(G) then G(L) denotes the subgraph of G such that $$V(G(L))=V(G)$$, $$E(G(L))=L$$. We call a graph almost regular if there is an integer d such that every vertex has valency d or $$d+1$$. Suppose that E(G) is partitioned into disjoint sets $$E_ 1,...,E_ r$$. A. J. W. Hilton [J. Comb. Theory, Ser. B 36, 125-134 (1984; Zbl 0542.05044)] found necessary and sufficient conditions for the existence of a detachment F of G such that F is a complete graph with $$2r+1$$ vertices and $$F(E_ i)$$ is a Hamilton circuit of F for $$i=1,...,r$$. We give a new proof of Hilton’s theorem, which also yields a generalization. Specifically, for any $$q\in \{0,1,...,r\}$$, we find necessary and sufficient conditions for G to have a detachment F without loops or multiple edges such that $$F(E_ 1),...,F(E_ r)$$ are almost regular and $$F(E_ 1,...,F(E_ q)$$ are 2-edge-connected and each vertex $$\xi$$ of G arises by identification from a prescribed number g($$\xi)$$ of vertices of F.

##### MSC:
 05C15 Coloring of graphs and hypergraphs 05C45 Eulerian and Hamiltonian graphs
##### Keywords:
vertex splitting; almost regular graphs; detachment
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##### References:
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