Complete tripartite subgraphs of balanced tripartite graphs with large minimum degree

TL;DR

This paper improves bounds on the minimum degree needed to guarantee the existence of a specific complete tripartite subgraph in balanced tripartite graphs, confirming a conjecture under certain adjacency conditions.

Contribution

It provides a tighter minimum degree bound for guaranteeing a complete tripartite subgraph and confirms a conjecture with an added adjacency assumption.

Findings

Improved the minimum degree bound to n+2n^{5/6} for octahedral subgraphs.

Confirmed the conjecture that n+cn^{1/2} suffices under additional adjacency conditions.

Established existence of such subgraphs with high minimum degree and adjacency constraints.

Abstract

In 1975 Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di asked what minimum degree guarantees an octahedral subgraph $K_3(2)$ in any tripartite graph $G$ with $n$ vertices in each vertex class. We show that $\delta(G)\geq n+2n^{\frac{5}{6}}$ suffices thus improving the bound $n+(1+o(1))n^{\frac{11}{12}}$ of Bhalkikar and Zhao obtained by following their approach. Bollob\'{a}s, Erd\H{o}s, and Szemer\'{e}di conjectured that $n+cn^{\frac{1}{2}}$ suffices and there are many $K_3(2)$-free tripartite graphs $G$ with $\delta(G)\geq n+cn^{\frac{1}{2}}$. We confirm this conjecture under the additional assumption that every vertex in $G$ is adjacent to at least $(1/5+\varepsilon)n$ vertices in any other vertex class.