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- W1676160334 abstract "It is shown that a hamiltonian n/2-regular bipartite graph G of order 2n > 8 contains a cycle of length 2n 2. Moreover, if such a cycle can be chosen to omit a pair of adjacent vertices, then G is bipancyclic. In (2), Entringer and Schmeichel gave a sufficient condition f or a hamiltonian bipartite graph to be bipancyclic. Theorem 1. A hamiltonian bipartite graph G of order 2n and size k Gk > n 2 /2 is bipancyclic (that is, contains cycles of all even lengths up to 2n). Interestingly enough, a non-hamiltonian graph with this same bound on the size may contain no long cycles whatsoever. Consider for instance, for n even, a graph obtained from the disjoint union of H1 = Kn/2,n/2 and H2 = Kn/2,n/2 by joining a single vertex of H1 with a vertex of H2. In the present note, we are interested in the cycle structure of a hamiltonian bipartite graph of order 2n, whose every vertex is of degree n/2. One immediately verifies that the size of such a graph is precisely n 2 /2, so the above theorem does not apply. Instead, we prove the following result. Theorem 2. If G is a hamiltonian n/2-regular bipartite graph of order 2n > 8, then G contains a cycle C of length 2n −2. Moreover, if C can be chosen to omit a pair of adjacent vertices, then G is bipancyclic." @default.
- W1676160334 created "2016-06-24" @default.
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- W1676160334 date "2007-11-28" @default.
- W1676160334 modified "2023-09-27" @default.
- W1676160334 title "On the cycle structure of hamiltonian k-regular bipartite graphs of order 4k" @default.
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