Abstract
E. Schmeichel and D. Hayes showed that ifG is a 2-connected graph withd(u) +d(v)≥n −1 for every pair of nonadjacent vertices andv, then G has a Hamiltonian cycle unlessG is the graph of Fig. 2 (b). In this paper, it is proved that, under almost the same conditions as Schmeichel and Hayes’s Theorem, namely,G is a 2-connected graph of ordern (n ≥ 40) with δ(G) ≥ 7 for every pair of nonadjacent vertices andv, G has two edge-disjoint Hamiltonian cycles unlessG is one of the graphs in Fig. 1 or Fig. 2, and this conclusion is best possible.
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Li, M. Two edge-disjoint hamiltonian cycles in graphs. Graphs and Combinatorics 10, 169–178 (1994). https://doi.org/10.1007/BF02986661
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DOI: https://doi.org/10.1007/BF02986661