1. Extremal totally regular mixed graphs and partially oriented incidence graphs of projective and biaffine planesTatiana Bagin Jajcay, Robert Jajcay, György Kiss, István Porupsánszki, 2025, original scientific article Abstract: An (r, z; g)-mixed graph is a graph containing both edges and darts satisfying the regularity property that each vertex of the graph is incident to r edges, z ingoing and z outgoing darts (called total regularity), and being of oriented girth g, i.e., containing an oriented cycle of length g, and no shorter oriented cycles. The problem addressed in this paper is analogous to the Cage Problem and calls for determining the orders of the smallest totally regular (r, z; g)-mixed graphs. We derive several upper and lower bounds on the orders of such minimal graphs, study the relations between these extremal graphs and their non-oriented or digraphical counterparts, and focus on properties of totally regular mixed graphs obtained by replacing some of the edges of the incidence graphs of projective and biaffine planes by darts. We also introduce two constructions based on introducing additional edges or darts into induced subgraphs of these incidence graphs.
Keywords: totally regular mixed graph, girth, projective plane, biaffine plane
Published in RUP: 04.06.2026; Views: 366; Downloads: 14
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2. On extremal (almost) edge-girth-regular graphsGabriela Araujo-Pardo, György Kiss, István Porupsánszki, 2025, original scientific article Abstract: 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.
Keywords: edge-girth-regular graph, cage problem, finite biaffine planes
Published in RUP: 03.11.2025; Views: 947; Downloads: 6
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4. A note on girth-diameter cagesGabriela Araujo-Pardo, Marston D. E. Conder, Natalia García-Colín, György Kiss, Dimitri Leemans, 2025, original scientific article Abstract: 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.
Keywords: cages, girth, degree-diameter problem
Published in RUP: 10.06.2025; Views: 1320; Downloads: 18
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