On regular induced subgraphs of generalized polygons

John Bamberg, Anurag Bishnoi, Gordon F. Royle

    Research output: Contribution to journalArticle

    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(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.

    Original languageEnglish
    Pages (from-to)254-275
    Number of pages22
    JournalJournal of Combinatorial Theory. Series A
    Volume158
    DOIs
    Publication statusPublished - 1 Aug 2018

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    Generalized Polygon
    Induced Subgraph
    Generalized Quadrangle
    Cage
    Graph in graph theory
    Regular Graph
    Subgraph
    Generalized Hexagon
    Lower bound
    Upper bound
    Expander
    Cayley
    Largest Eigenvalue
    Girth
    Projective plane
    Automorphism
    Hexagon
    Bipartite Graph
    Lemma
    Incidence

    Cite this

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    title = "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 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.",
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    author = "John Bamberg and Anurag Bishnoi and Royle, {Gordon F.}",
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    On regular induced subgraphs of generalized polygons. / Bamberg, John; Bishnoi, Anurag; Royle, Gordon F.

    In: Journal of Combinatorial Theory. Series A, Vol. 158, 01.08.2018, p. 254-275.

    Research output: Contribution to journalArticle

    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

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