### 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 subgraph 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 that c(q,8)⩽2(q^{3}−qq−q) for 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.

Original language | English |
---|---|

Pages (from-to) | 254-275 |

Number of pages | 22 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 158 |

DOIs | |

Publication status | Published - 1 Aug 2018 |

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*Journal of Combinatorial Theory. Series A*, vol. 158, pp. 254-275. https://doi.org/10.1016/j.jcta.2018.03.015

**On regular induced subgraphs of generalized polygons.** / Bamberg, John; Bishnoi, Anurag; Royle, Gordon F.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On regular induced subgraphs of generalized polygons

AU - Bamberg, John

AU - Bishnoi, Anurag

AU - Royle, Gordon F.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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 subgraph 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 that c(q,8)⩽2(q3−qq−q) for q an even power of a prime, and c(q,12)⩽2(q5−3q3) 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.

AB - 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 subgraph 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 that c(q,8)⩽2(q3−qq−q) for q an even power of a prime, and c(q,12)⩽2(q5−3q3) 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.

KW - Cage

KW - Expander mixing lemma

KW - Generalized polygon

KW - Moore graph

KW - t-Good structure

UR - http://www.scopus.com/inward/record.url?scp=85044139584&partnerID=8YFLogxK

UR - https://arxiv.org/abs/1708.01095

U2 - 10.1016/j.jcta.2018.03.015

DO - 10.1016/j.jcta.2018.03.015

M3 - Article

VL - 158

SP - 254

EP - 275

JO - Journal of Combinatorial Theory Series A

JF - Journal of Combinatorial Theory Series A

SN - 0021-9800

ER -