Abstract
For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let π(t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let γ(t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that π(t,n)≥A·log(n) and γ(t,n)·log(γ(t,n))≥B·log(n). Examples are presented showing that fort≤1 there exist constantsA′, B′ such that π(t,n)≤A′·log(n) and γ(t,n)≤B′· log(n). It is conjectured that γ(t,n) ≥B″·log(n) for some constantB″. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK 1.l-free graphs, wherel is fixed.
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Broersma, H.J., van den Heuvel, J., Jung, H.A. et al. Long paths and cycles in tough graphs. Graphs and Combinatorics 9, 3–17 (1993). https://doi.org/10.1007/BF01195323
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DOI: https://doi.org/10.1007/BF01195323