10 Problems for Partitions of Triangle-free Graphs

TL;DR

This paper explores partition problems in triangle-free graphs related to Erdős' Sparse Half Conjecture, proving a sharp bound for large graphs and discussing similar problems for $K_4$-free graphs.

Contribution

It proves a variant of Erdős' Sparse Half Conjecture for large triangle-free graphs with a sharp bound and discusses related problems for $K_4$-free graphs.

Findings

Proved a sharp partition bound for large triangle-free graphs.

Established a specific partition with balanced edges.

Discussed analogous problems for $K_4$-free graphs.

Abstract

We will state 10 problems, and solve some of them, for partitions in triangle-free graphs related to Erd\H{o}s' Sparse Half Conjecture. Among others we prove the following variant of it: For every sufficiently large even integer $n$ the following holds. Every triangle-free graph on $n$ vertices has a partition $V(G)=A\cup B$ with $|A|=|B|=n/2$ such that $e(G[A])+e(G[B])\leq n^2/16$. This result is sharp since the complete bipartite graph with class sizes $3n/4$ and $n/4$ achieves equality, when $n$ is a multiple of 4. Additionally, we discuss similar problems for $K_4$-free graphs.