On subgraphs of tripartite graphs

TL;DR

This paper examines the minimum degree conditions in tripartite graphs that guarantee the presence of an octahedral subgraph, refining previous bounds and providing constructions that challenge existing conjectures.

Contribution

It demonstrates that earlier methods only yield a weaker bound and offers new constructions suggesting a tighter bound may be achievable.

Findings

Previous bounds are weaker than conjectured.

New constructions challenge existing minimum degree conditions.

Refined bounds for tripartite graph substructure presence.

Abstract

Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di [Discrete Math 13 (1975), 97--107] investigated a tripartite generalization of the Zarankiewicz problem: what minimum degree forces a tripartite graph with $n$ vertices in each part to contain an octahedral graph $K_3(2)$? They proved that $n+2^{-1/2}n^{3/4}$ suffices and suggested it could be weakened to $n+cn^{1/2}$ for some constant $c>0$. In this note we show that their method only gives $n+ (1+o(1)) n^{11/12}$ and provide many constructions that show if true, $n+ c n^{1/2}$ is better possible.