Characterizing Circular Colouring Mixing for $\frac{p}{q}<4$

TL;DR

This paper characterizes the conditions under which graphs are $(p,q)$-mixing for $rac{p}{q}<4$, establishes complexity results, and provides algorithms for specific graph classes, advancing understanding of circular coloring reconfiguration.

Contribution

It provides a complete characterization of $(p,q)$-mixing for $rac{p}{q}<4$, proves co-NP-completeness for certain cases, and offers a polynomial algorithm for planar graphs.

Findings

$(p,q)$-mixing characterized by cycle wind conditions

$(2k+1,k)$-mixing is co-NP-complete

Circular mixing number of bipartite graphs is 2

Abstract

Given a graph $G$, the $k$-mixing problem asks: Can one obtain all $k$-colourings of $G$, starting from one $k$-colouring $f$, by changing the colour of only one vertex at a time, while at each step maintaining a $k$-colouring? More generally, for a graph $H$, the $H$-mixing problem asks: Can one obtain all homomorphisms $G \to H$, starting from one homomorphism $f$, by changing the image of only one vertex at a time, while at each step maintaining a homomorphism $G \to H$? This paper focuses on a generalization of $k$-colourings, namely $(p,q)$-circular colourings. We show that when $2 < \frac{p}{q} < 4$, a graph $G$ is $(p,q)$-mixing if and only if for any $(p,q)$-colouring $f$ of $G$, and any cycle $C$ of $G$, the wind of the cycle under the colouring equals a particular value (which intuitively corresponds to having no wind). As a consequence we show that $(p,q)$-mixing is closed under a restricted homomorphism called a fold. Using this, we deduce that $(2k+1,k)$-mixing is co-NP-complete for all $k \in \mathbb{N}$, and by similar ideas we show that if the circular chromatic number of a connected graph $G$ is $\frac{2k+1}{k}$, then $G$ folds to $C_{2k+1}$. We use the characterization to settle a conjecture of Brewster and Noel, specifically that the circular mixing number of bipartite graphs is $2$. Lastly, we give a polynomial time algorithm for $(p,q)$-mixing in planar graphs when $3 \leq \frac{p}{q} <4$.