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A note on girth-diameter cages

par Araujo-Pardo, Gabriela;Conder, Marston;Garcia Colin, Natalia ;Kiss, Gyorgy;Leemans, Dimitri
Référence The art of discrete and applied mathematics, 8, 3.06
Publication Publié, 2025-07-01

Article révisé par les pairs

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Résumé :

In this paper we introduce a problem closely related to the Cage Problem and the Degree Diameter Problem. For integers k ≥ 2, g ≥ 3 and d ≥ 1, we define a (k; g, d)-graph to be a k-regular graph with girth g and diameter d. We denote by n0(k; g, d) the smallest possible order of such a graph, and, if such a graph exists, we call it a (k; g, d)-cage. In particular, we focus on (k; 5, 4)-graphs. We show that n0(k; 5, 4) ≥ k2 + k + 2 for all k, and report on the determination of all (k; 5, 4)-cages for k = 3, 4 and 5 and of examples with k = 6, and describe some examples of (k; 5, 4)-graphs which prove that n0(k; 5, 4) ≤ 2k2 for infinitely many k.