On Alternating 6-Cycles in Edge-Coloured Graphs

TL;DR

This paper proves that in large red/blue edge-coloured complete graphs, the maximum number of alternating 6-cycles is achieved by a random colouring, resolving a previously open problem.

Contribution

It establishes the asymptotic maximum of alternating 6-cycles in large edge-coloured cliques using flag algebra methods, settling an open problem.

Findings

Random colouring asymptotically maximizes alternating 6-cycles.

Uses flag algebra technique for combinatorial optimization.

Solves the first open case of the problem by Basit et al.

Abstract

In this short note, we use flag algebras to prove that the number of colour alternating 6-cycles in a red/blue colouring of a large clique is asymptotically maximized by a uniformly random colouring. This settles the first open case of a problem of Basit, Granet, Horsley, K\"undgen and Staden.