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Regular \(n\)-valent \(n\)-connected non-Hamiltonian non \(n\)-edge-colourable graphs. (English) Zbl 0237.05106
For all \(n\geq 3\), except \(n=5,6,\) or 7 regular \(n\)-valent non-Hamiltonian non \(n\)-edge-colourable graphs are constructed by replacing edges of Petersen’s graph by multiple edges, then replacing vertices with complete bipartite graphs. For \(n=5,6\), or 7, the same procedure gives graphs that are regular \(n\)-valent, Hamiltonian, and non \(n\)-edge-colourable but not \(n\)-connected.
Reviewer: G. H. J. Meredith

MSC:
05C15 Coloring of graphs and hypergraphs
05C35 Extremal problems in graph theory
05C99 Graph theory
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[1] {\scG. H. J. Meredith}, Some families of non-Hamiltonian graphs, Proc. 1972 Oxford Combinatorics Conf., (D. J. A. Welsh, Ed.), I. M. A., to appear. · Zbl 0237.05106
[2] Nash-Williams, C.St.J.A., Possible directions in graph theory, () · Zbl 0263.05101
[3] Vizing, V.G., On an estimate of the chromatic class of a p-graph, Disker. analiz., 3, 25-30, (1964)
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