Short and local transformations between ($\Delta+1$)-colorings

TL;DR

This paper improves the understanding of recoloring sequences between ($ abla+1$)-colorings of graphs with maximum degree $ abla$, showing that such sequences can be linear in length and locally computed in distributed settings.

Contribution

The authors prove that recoloring sequences between non-frozen ($ abla+1$)-colorings can be linear in length, and they develop a local, distributed algorithm for efficient recoloring.

Findings

Existence of linear-length recoloring sequences for non-frozen colorings.

Development of a local, distributed recoloring algorithm.

Improved bounds over previous quadratic results.

Abstract

Recoloring a graph is about finding a sequence of proper colorings of this graph from an initial coloring $\sigma$ to a target coloring $\eta$. Adding the constraint that each pair of consecutive colorings must differ on exactly one vertex, one asks: Is there a sequence of colorings from $\sigma$ to $\eta$? If yes, how short can it be? In this paper, we focus on $(\Delta+1)$-colorings of graphs of maximum degree $\Delta$. Feghali, Johnson and Paulusma proved that, if both colorings are non-frozen (i.e. we can change the color of a least one vertex), then a quadratic recoloring sequence always exists. We improve their result by proving that there actually exists a linear transformation (assuming that $\Delta$ is a constant). In addition, we prove that the core of our algorithm can be performed locally. Informally, this means that after some preprocessing, the color changes that a given node has to perform only depend on the colors of the vertices in a constant size neighborhood. We make this precise by designing of an efficient recoloring algorithm in the LOCAL model of distributed computing.