New families of graphs without short cycles and large size

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Abstract

We denote by e x (n ; {C3, C4, ..., Cs}) or fs (n) the maximum number of edges in a graph of order n and girth at least s + 1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g - 1 on fewer vertices. For an infinite sequence of values of n and s ∈ {4, 6, 10} the obtained graphs are denser than the known constructions of graphs of the same girth s + 1. We also give another different construction of dense graphs for an infinite sequence of values of n and s ∈ {7, 11}. These two methods improve the known lower bounds on fs (n) for s ∈ {4, 6, 7, 10, 11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that lim supn → ∞ frac(fs (n), n1 + frac(2, s - 1)) = 2- 1 - frac(2, s - 1) for s ∈ {5, 7, 11}, and s- 1 - frac(2, s) ≤ lim supn → ∞ frac(fs (n), n1 + frac(2, s)) ≤ 0.5 for s ∈ {6, 10}. © 2010 Elsevier B.V. All rights reserved.

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Abajo, E., Balbuena, C., & Diánez, A. (2010). New families of graphs without short cycles and large size. Discrete Applied Mathematics, 158(11), 1127–1135. https://doi.org/10.1016/j.dam.2010.03.007

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