Abstract
Given a fixed positive integer k ≥ 2, let G be a simple graph of order n ≥ 6k. It is proved that if the minimum degree of G is at least n/2 + 1, then for every pair of vertices x and y, there exists a Hamiltonian cycle such that the distance between x and y along that cycle is precisely k.
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References
Chartrand G., Lesniak L.: Graphs and Digraphs. Chapman and Hall, London (2005)
Dirac G.A.: Some theorems on abstract graphs. Proc. London Math. Soc. 2, 69–81 (1952)
Faudree R., Gould R., Jacobson M., Magnant C.: Distributing vertices on Hamiltonian cycles. J. Graph Theory 69, 28–45 (2012)
Faudree, R., Li, H.: Locating pairs of vertices on a Hamiltonian cycle. Discrete Math. 312, 2707–2719 (2013)
Kaneko A., Yoshimoto K.: On a Hamiltonian cycle in which specified vertices are uniformly distributed. J. Combin. Theory Ser. B 81, 309–324 (2001)
Moon J., Moser L.: On hamiltonian bipartite graphs. Israel J. Math. 1, 163–165 (1963)
St, C., Nash-Williams, J.A.: Edge-disjoint Hamiltonian circuits in graphs with vertices of large valency, in Studies in Pure Mathematics, pp. 157–183. Academic Press, London (1971)
Sárközy G., Selkow S.: Distributing vertices along a Hamiltonian cycle in Dirac graphs. Discrete Math. 308, 5757–5770 (2008)
Whitney H.: Congruent graphs and the connectivity of graphs. Am. J. Math. 54, 150–168 (1932)
Williamson J.: Panconnected graphs. II. Period. Math. Hungar. 8, 105–116 (1977)
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Faudree, R.J., Lehel, J. & Yoshimoto, K. Note on Locating Pairs of Vertices on Hamiltonian Cycles. Graphs and Combinatorics 30, 887–894 (2014). https://doi.org/10.1007/s00373-013-1325-9
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DOI: https://doi.org/10.1007/s00373-013-1325-9