Discrete Mathematics & Theoretical Computer Science, Vol 15, No 1 (2013)

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Further results on Maximal Nontraceable graphs of smallest size

Joy Elizabeth Singleton, Alewyn Petrus Burger


Let g(n) denote the minimum number of edges of a maximal nontraceable (MNT) graph of order n. In 2005 Frick and Singleton (Lower bound for the size of maximal nontraceable graphs, Electronic Journal of Combinatorics, 12(1) R32, 2005) proved that g(n) = ⌈3n-22 ⌉ for n ≥54 as well as for n ∈I, where I= {12,13,22,23,30,31,38,39, 40,41,42,43,46,47,48,49,50,51} and they determined g(n) for n ≤9. We determine g(n) for 18 of the remaining 26 values of n, showing that g(n) = ⌈ 3n-22 ⌉ for n ≥54 as well as for n ∈I ∪{18,19,20,21,24,25,26,27,28, 29,32,33} and g(n) = ⌈ 3n2 ⌉ for n ∈{ 10, 11, 14, 15, 16, 17}. We give results based on ``analytic'' proofs as well as computer searches.

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