Forbidden subgraphs in the norm graph

TL;DR

This paper investigates the properties of the norm graph, demonstrating that it contains no certain complete bipartite subgraphs, specifically showing the absence of a larger bipartite graph than previously known.

Contribution

It proves that the norm graph, which avoids a specific bipartite subgraph, also does not contain a larger bipartite subgraph of a related form, refining understanding of its structure.

Findings

Norm graph contains no copy of K_{t+1,(t-1)!-1}

Norm graph has about 0.5 n^{2-1/t} edges

Norm graph avoids certain large bipartite subgraphs

Abstract

We show that the norm graph constructed in [J. Koll\'{a}r, L. R\'{o}nyai and T. Szab\'o, Norm-graphs and bipartite Tur\'{a}n numbers, Combinatorica, 16 (1996) 399--406] with $n$ vertices about $\frac{1}{2}n^{2-1/t}$ edges, which contains no copy of $K_{t,(t-1)!+1}$, does not contain a copy of $K_{t+1,(t-1)!-1}$.