Non-monochromatic Triangles in a 2-Edge-Coloured Graph

TL;DR

This paper proves a condition under which a 2-edge-coloured graph contains a triangle with edges of both colours, contributing to understanding edge-colouring properties and triangle existence.

Contribution

It establishes a new sufficient condition for the existence of non-monochromatic triangles in 2-edge-coloured graphs.

Findings

Proves a threshold condition involving edge counts for non-monochromatic triangles.

Identifies a relationship between total edges, colour class sizes, and triangle existence.

Proposes a conjecture for generalization to multiple colour partitions.

Abstract

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$. We conjecture a natural generalization to partitions with more blocks.