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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: 350; Downloads: 2
Full text (547,76 KB)

Abstract: The CD(n,q) graphs are connected components of q-regular graphs D(n,q) introduced in 1995 by Lazebnik and Ustimenko. They constitute the best universal family of regular graphs of prime power degree with regard to the Cage Problem which calls for determining the orders of the smallest k-regular graphs of girth g. The girths of the CD(n,q) graphs are known to be at least n+4 in case of even n, and n+5 for odd n. We propose to extend the use of the CD(n,q) graphs into the area of mixed graphs by adding directions to certain edges of the C(n,q)graphs. In the context of mixed graphs, graphs in which the number of incident non-oriented edges is the same for all vertices, and the numbers of out-going and in-going edges are also equal and the same for all vertices, are of special interest and are called totally regular mixed graphs. In view of the special properties of the original C(n,q) graphs with regard to cages, we believe that the totally regular mixed graphs we propose to study may also prove to be extremal with regard to properties sought for in the area of mixed graphs.
Keywords: cage problem, girth, degree, mixed graphs
Published in RUP: 03.11.2025; Views: 302; Downloads: 2
Full text (574,95 KB)