Flip colouring of graphs II

TL;DR

This paper investigates flip colourings in graphs, establishing conditions for their existence and constructing specific examples, while also exploring sequences of flip parameters with particular growth properties.

Contribution

It provides new existence results for (b,r)-flip graphs under certain bounds and analyzes the behavior of flip sequences over multiple steps.

Findings

Existence of (b,r)-flip graphs for certain b, r within specified bounds.

Construction of flip graphs with size proportional to b+r.

Analysis of flip sequences with unbounded growth over multiple steps.

Abstract

We give results concerning two problems on the recently introduced \textit{flip colourings of graphs}. For positive integers $b, r$ with $b < r$, we say that a $b + r$ regular graph is a $(b,r)$-\textit{flip graph} if there exists a red/blue edge colouring such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed neighbourhood of every vertex there are more blue edges than red edges. We prove that for integers $b, r$ with $4 \leq b < r < b + 2 \left\lfloor\frac{b+2}{6}\right\rfloor^2$, small constructions of $(b,r)$-flip graphs on $\Theta(b+r)$ vertices are possible. Furthermore, we prove that there exist $k$-flip sequences $(a_1, \dots, a_k)$ where $k > 4$, such that $a_k$ can be arbitrarily large whilst $a_i$ is constant for $1 \leq i < \frac{k}{4}$.