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
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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 log(2) n + 2 where c = (log(2)3-1)(-1) approximate to 1.71 thereby improving a bound given by Bondy and Jackson [3].