Burger, Alewyn Petrus and Singleton, Joy Elizabeth - Further results on maximal nontraceable graphs of smallest size

dmtcs:627 - Discrete Mathematics & Theoretical Computer Science, February 28, 2013, Vol. 15 no. 1
Further results on maximal nontraceable graphs of smallest size

Authors: Burger, Alewyn Petrus and Singleton, Joy Elizabeth

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.


Source : oai:HAL:hal-00990607v1
Volume: Vol. 15 no. 1
Section: Graph Theory
Published on: February 28, 2013
Submitted on: April 16, 2012
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]


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