Exploring Projective Norm Graphs

Abstract

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).

Generalized Turán problems for complete bipartite graphs

“…Bayer, Mészáros, Rónyai and Szabó [3] also showed ex(n, K a,b , K s,t ) = Θ(n a+b−ab/s ) in case a ≤ 3, a ≤ b < s, s ≥ 4 and t ≥ (s−1)! +1.…”

“…Bayer, Mészáros, Rónyai and Szabó [3] showed ex(n, K 4,6 , K 4,7 ) = Ω(n 7/4 ). For other values of s, we again use Theorem 1.3.…”

“…The order of magnitude of ex(n, K a,b , K s,t ) was studied by Alon and Shikhelman [2] in case a ≤ s ≤ t and a ≤ b < t. Their results were extended by Ma, Yuan and Zhang [24] in the case a < s, b ≤ s and t is large enough and by Bayer, Mészáros, Rónyai and Szabó [3] in the case a ≤ 3, s ≥ 4, a ≤ b < t and t is large enough. Gerbner, Methuku and Vizer [12] studied the case a, s ≤ b ≤ t. Gerbner [10] determined ex(n, K 1,b , K 2,2 ) exactly.…”

Singer difference sets and the projective norm graph

“…Furthermore, the proof also provided many copies of K 4,6 . Computer calculations have also suggested that the same holds in the case p ∈ {2, 3} as well, but the arguments in [2] crucially used the restriction on the characteristic. In this paper we provide different arguments to show the existence of K 4,6 in N G(q, 4) for the cases when p ≡ 1 (mod 3) and q ≥ 5, and hence establish s(4) = 6.…”

“…In this direction Grosu [14] has recently shown that NG(p, 4) contains a copy of the complete bipartite graph K 4,6 for roughly 1 9 -fraction of all primes p. In [2], among other things, this was extended for any prime power q if the characteristic p is not 2 or 3. Furthermore, the proof also provided many copies of K 4,6 .…”