Cycles through ten vertices in 3-connected cubic graphs


It is known that there exists a cycle through any nine vertices of a 3-connected cubic graphG. Here we show that if an edge is removed from such a graph, then there is still a cycle through any five vertices. Furthermore, we characterise the circumstances in which there fails to be a cycle through six. As corollaries we are able to prove that a 3-connected cubic graph has a cycle through any specified five vertices and one edge, and to classify the conditions under which it has a cycle through four chosen vertices and two edges.

We are able to use the five and six vertex results to show that a 3-connected cubic graph has a cycle which passes through any ten given vertices if and only if the graph is not contractible to the Petersen graph in such a way that the ten vertices each map to a distinct vertex of the Petersen graph.

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Ellingham, M.N., Holton, D.A. & Little, C.H.C. Cycles through ten vertices in 3-connected cubic graphs. Combinatorica 4, 265–273 (1984).

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AMS subject classification (1980)

  • 05 C 40
  • 05 C 38