Abstract
A graph, G, is called uniquely Hamiltonian if it contains exactly one Hamilton cycle. We show that if G=(V, E) is uniquely Hamiltonian then
Where #(G)=1 if G has even number of vertices and 2 if G has odd number of vertices. It follows that every n-vertex uniquely Hamiltonian graph contains a vertex whose degree is at most c log2n+2 where c=(log23−1)−1≈1.71 thereby improving a bound given by Bondy and Jackson [3].
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Abbasi, S., Jamshed, A. A Degree Constraint for Uniquely Hamiltonian Graphs. Graphs and Combinatorics 22, 433–442 (2006). https://doi.org/10.1007/s00373-006-0666-z
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DOI: https://doi.org/10.1007/s00373-006-0666-z