On regular induced subgraphs of generalized polygons

Abstract

The cage problem asks for the smallest number c(k, g) of vertices in a k-regular graph of girth g and graphs meeting this bound are known as cages. While cages are known to exist for all integers k 2 and g 3, the exact value of c(k, g) is known only for some small values of k, g and three infinite families where g ∈ {6, 8, 12} and k − 1 is a prime power. These infinite families come from the incidence graphs of generalized polygons. Some of the best known upper bounds on c(k, g) for g ∈ {6, 8, 12} have been obtained by constructing small regular induced subgraphs of these cages.In this paper, we first use the Expander Mixing Lemma to give a general lower bound on the size of an induced k-regular subgraph of a regular bipartite graph in terms of the second largest eigenvalue of the host graph. We use this bound to show that the known construction of (k, 6)-graphs using Baer subplanes of the Desarguesian projective plane is the best possible. For generalized quadrangles and hexagons, our bounds are new. In particular, we improve the known lower bound on the size of an induced q-regular subgraphs of the classical generalized quadrangle Q(4, q) and show that the known constructions are asymptotically sharp, which answers a question of Metsch [21, Section 6].For prime powers q, we also improve the known upper bounds on c(q, 8) and c(q, 12) by giving new geometric constructions of q-regular induced subgraphs in the symplectic generalized quadrangle W(3, q) and the split Cayley hexagon H(q), respectively. Our constructions show thatfor q an even power of a prime, and c(q, 12) 2(q 5 − 3q 3 ) for all prime powers q. For q ∈ {3, 4, 5} we also give a computer classification of all q-regular induced subgraphs of the classical generalized quadrangles of order q. For W(3, 7) we classify all 7-regular induced subgraphs which have a non-trivial automorphism.2010 Mathematics Subject Classification. Primary 05C50, 05C35, 51E12.

Bipartite biregular Moore graphs

“…In the first part of this section, we recall the connection between Moore graphs and generalized polygons that was extensively studied (see, for instance, Bamberg, Bishnoi, and Royle [2]) because we will use it for the rest of the paper. In fact, our first result is an immediate consequence of the result proved by Araujo-Pardo, Jajcay, and Ramos in [1], and the analysis given in the introduction about the coincidence of the bounds for bipartite biregular cages and bipartite biregular Moore graphs when d is even.…”

“…As already mentioned, G is bipartite, has 2r 2 − 2r + 1 vertices, and diameter 3. Then, by Theorem 4.3, the semi-double graph G [1] (or G [2] ) is a bipartite graph on 3r 2 − 3r + 3 vertices, biregular with degrees r and 2r, and diameter 3. Thinking on the projective plane, this corresponds to duplicate, for instance, each line, so that each point is on 2r lines, and each line has r points, as before.…”

On the automorphisms of a family of small q-regular graphs of girth 8

“…By the nature of the construction, these graphs are bipartite and highly symmetric. There were several attempts to construct (k, g)cages, or at least small (k, g)-graphs from these graphs as induced subgraphs (e.g., see [1,2,3,4,5]), when k − 1 is not a prime power and g ∈ {6, 8, 12}, because in these cases no (k, g)-cages are known. Trivially, the induced subgraphs are also necessarily bipartite, so there is some interest to investigate, whether these graphs are vertex-transitive, or at least transitive on both parts, in the sense, that the two parts are formed by two fibers of a graph obtained as a lift from a dipole.…”

“…As we already mentioned, the graph I(q) (or Γ q in our notation) served as an initial graph in many constructions of (k, 8)-graphs for k < q + 1. See, for example [1,4,5,11].…”