1. On (r,g,χ)- graphs and cages of regularity r, girth g and chromatic number χGabriela Araujo-Pardo, Julio César Díaz-Calderón, Julián Fresán-Figueroa, Diego González-Moreno, Linda Lesniak, Mika Olsen, 2025, izvirni znanstveni članek Opis: For integers r ≥ 2, g ≥ 3 and χ ≥ 2, an (r, g, χ)-graph is an r-regular graph with girth g and chromatic number χ. Such a graph of minimum order is called an (r, g, χ)-cage. Here we prove the existence of (r, g, χ)-graphs for all r and even g when χ = 2 and for all r and g when χ = 3. Furthermore, using both existence proofs and explicit constructions we give examples of (r, g, χ)-graphs for infinitely many values of r, g, χ.
Ključne besede: graphs, cages, girth, chromatic number
Objavljeno v RUP: 03.11.2025; Ogledov: 117; Prenosov: 2
 Celotno besedilo (408,46 KB) |
2. On extremal (almost) edge-girth-regular graphsGabriela Araujo-Pardo, György Kiss, István Porupsánszki, 2025, izvirni znanstveni članek Opis: A k-regular graph of girth g is called an edge-girth-regular graph, or an egr-graph for short, if each of its edges is contained in exactly λ distinct g-cycles. An egr-graph is called extremal for the triple (k, g, λ) if has the smallest possible order. We prove that some graphs arising from incidence graphs of finite planes are extremal egr-graphs. We also prove new lower bounds on the order of egr-graphs.
Ključne besede: edge-girth-regular graph, cage problem, finite biaffine planes
Objavljeno v RUP: 03.11.2025; Ogledov: 136; Prenosov: 1
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3. A note on girth-diameter cagesGabriela Araujo-Pardo, Marston D. E. Conder, Natalia García-Colín, György Kiss, Dimitri Leemans, 2025, izvirni znanstveni članek Opis: 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.
Ključne besede: cages, girth, degree-diameter problem
Objavljeno v RUP: 10.06.2025; Ogledov: 577; Prenosov: 15
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