Forbidden subgraphs in the norm graph

Ball, Simeon; Pepe, Valentina

doi:10.1016/j.disc.2015.11.010

Cited by 2 publications

(4 citation statements)

References 10 publications

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“…In this direction, Ball and Pepe [11,12] recently proved that the K t,(t−1)!+1 -free projective norm graphs do not contain a K t+1,(t−1)!−1 , which in particular improved the earlier probabilistic lower bound on ex(n, K 5,5 ).…”

Section: The Projective Norm Graphsmentioning

confidence: 86%

“…Since their first appearance, projective norm graphs were studied extensively [7,11,12,30,34,47,52]. Their various properties were utilized in many other areas, both within and outside combinatorics.…”

Section: The Projective Norm Graphsmentioning

confidence: 99%

“…Before solving these systems, we show that their solution sets are disjoint, i.e., the number of solutions to (9) is the sum of the number of solutions to ( 14), ( 15), ( 16) and (17). Indeed, assume that there was a solution X ∈ F q 3 of (9), satisfying both equations in (12). Then we have…”

Section: Finding a K 46mentioning

confidence: 99%

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Exploring Projective Norm Graphs

Bayer¹,

Mészáros²,

Rónyai³

et al. 2019

Preprint

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The projective norm graphs NG(q, t) provide tight constructions for the Turán number of complete bipartite graphs K t,s with s > (t − 1)!. In this paper we determine their automorphism group and explore their small subgraphs. To this end we give quite precise estimates on the number of solutions of certain equation systems involving norms over finite fields. The determination of the largest integer s t , such that the projective norm graph NG(q, t) contains K t,st for all large enough prime powers q is an important open question with far-reaching general consequences. The best known bounds, t − 1 ≤ s t ≤ (t − 1)!, are far apart for t ≥ 4. Here we prove that NG(q, 4) does contain (many) K 4,6 for any prime power q not divisble by 2 or 3. This greatly extends recent work of Grosu, using a completely different approach. Along the way we also count the copies of any fixed 3-degenerate subgraph, and find that projective norm graphs are quasirandom with respect to this parameter. Some of these results also extend the work of Alon and Shikhelman on generalized Turán numbers. Finally we also give a new, more elementary proof for the K 4,7 -freeness of NG(q, 4).

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Section: The Projective Norm Graphsmentioning

confidence: 86%

Section: The Projective Norm Graphsmentioning

confidence: 99%

See 1 more Smart Citation

Exploring Projective Norm Graphs

Bayer¹,

Mészáros²,

Rónyai³

et al. 2019

Preprint

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show abstract

“…Since their first appearance, (projective) norm graphs were studied extensively [1,9,10,20,23,34,39]. Their various properties were utilized in many other areas, both within and outside combinatorics.…”

Section: The Automorphism Groupmentioning

confidence: 99%

The automorphism group of projective norm graphs

Bayer

Mészáros

Rónyai

et al. 2023

AAECC

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The projective norm graphs are central objects to extremal combinatorics. They appear in a variety of contexts, most importantly they provide tight constructions for the Turán number of complete bipartite graphs $$K_{t,s}$$ K t , s with $$s>(t-1)!$$ s > ( t - 1 ) ! . In this note we deepen their understanding further by determining their automorphism group.

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Forbidden subgraphs in the norm graph

Abstract: We show that the norm graph constructed in [11] with n vertices about 1 2 n 2−1/t edges, which contains no copy of K t,(t−1)!+1 , does not contain a copy of K t+1,(t−1)!−1 .

Cited by 2 publications

References 10 publications

Exploring Projective Norm Graphs

Exploring Projective Norm Graphs

The automorphism group of projective norm graphs

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