# A polynomial version of Cereceda's conjecture

## TL;DR

This paper proves that the diameter of the reconfiguration graph for proper colorings of a $d$-degenerate graph is polynomially bounded, specifically $O(n^{d+1})$, and confirms Cereceda's conjecture for planar bipartite graphs.

## Contribution

It establishes polynomial bounds on the diameter of reconfiguration graphs for $d$-degenerate graphs and confirms the conjecture for planar bipartite graphs.

## Key findings

- Diameter of $k$-reconfiguration graph is $O(n^{d+1})$ for $k
geq d+2$.
- Quadratic diameter for $k
geq 1.5(d+1)$.
- Quadratic diameter for planar bipartite graphs.

## Abstract

Let $k$ and $d$ be such that $k \ge d+2$. Consider two $k$-colourings of a $d$-degenerate graph $G$. Can we transform one into the other by recolouring one vertex at each step while maintaining a proper coloring at any step? Cereceda et al. answered that question in the affirmative, and exhibited a recolouring sequence of exponential length. However, Cereceda conjectured that there should exist one of quadratic length. The $k$-reconfiguration graph of $G$ is the graph whose vertices are the proper $k$-colourings of $G$, with an edge between two colourings if they differ on exactly one vertex. Cereceda's conjecture can be reformulated as follows: the diameter of the $(d+2)$-reconfiguration graph of any $d$-degenerate graph on $n$ vertices is $O(n^2)$. So far, the existence of a polynomial diameter is open even for $d=2$. In this paper, we prove that the diameter of the $k$-reconfiguration graph of a $d$-degenerate graph is $O(n^{d+1})$ for $k \ge d+2$. Moreover, we prove that if $k \ge \frac 32 (d+1)$ then the diameter of the $k$-reconfiguration graph is quadratic, improving the previous bound of $k \ge 2d+1$. We also show that the $5$-reconfiguration graph of planar bipartite graphs has quadratic diameter, confirming Cereceda's conjecture for this class of graphs.