Maximizing Alternating Paths via Entropy

TL;DR

This paper establishes an upper bound on the number of specific alternating edge walks in edge-colored graphs, solving a recent open problem using entropy-based techniques.

Contribution

It provides the first tight upper bound on alternating walks of a given length in red-blue edge-colored graphs, employing the entropy method.

Findings

Proves an explicit upper bound formula for alternating walks.

Solves a recent open problem in graph theory.

Introduces entropy method to combinatorial enumeration.

Abstract

We prove that if $G$ is an $n$-vertex graph whose edges are coloured with red and blue, then the number of colour-alternating walks of length $2k+1$ with $k+1$ red edges and $k$ blue edges is at most $k^k(k+1)^{k+1}(2k+1)^{-2k-1}n^{2k+2}$. This solves a problem that was recently posed by Basit, Granet, Horsley, K\"undgen and Staden. Our proof involves an application of the entropy method.