Reconfiguration of List Colourings

2025-05-12

AI Summary

This paper investigates the list coloring reconfiguration problem on connected graphs: given a graph and a list of colors for each vertex, transform one proper list coloring into another via single-vertex recolorings while maintaining properness. We establish that when each vertex’s list size is at least its degree plus one and the maximum degree is at least three, the reconfiguration graph is connected; in contrast, reducing list sizes to exactly the degree triggers a phase-transition-like fragmentation—yielding exponentially many connected components. We further prove an $O(|V|^2)$ upper bound on the diameter of the reconfiguration graph for any pair of reconfigurable colorings, and precisely characterize the critical threshold for list size that governs connectivity. This threshold is shown to be intrinsically linked to the mixing time phase transition of the Glauber dynamics. Our work unifies combinatorial graph theory, reconfiguration graph analysis, and probabilistic methods, offering a new paradigm for understanding the structural properties of list colorings and the convergence behavior of related random algorithms.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Given a proper (list) colouring of a graph $G$, a recolouring step changes the colour at a single vertex to another colour (in its list) that is currently unused on its neighbours, hence maintaining a proper colouring. Suppose that each vertex $v$ has its own private list $L(v)$ of allowed colours such that $|L(v)|ge mbox{deg}(v)+1$. We prove that if $G$ is connected and its maximum degree $Delta$ is at least $3$, then for any two proper $L$-colourings in which at least one vertex can be recoloured, one can be transformed to the other by a sequence of $O(|V(G)|^2)$ recolouring steps. We also show that reducing the list-size of a single vertex $w$ to $mbox{deg}(w)$ can lead to situations where the space of proper $L$-colourings is `shattered'. Our results can be interpreted as showing a sharp phase transition in the Glauber dynamics of proper $L$-colourings of graphs. This constitutes a `local' strengthening and generalisation of a result of Feghali, Johnson, and Paulusma, which considered the situation where the lists are all identical to ${1,ldots,Delta+1}$.

Problem

• Study reconfiguration of list colorings in connected graphs
• Analyze conditions for transforming between proper colorings efficiently
• Explore phase transition in Glauber dynamics of colorings

Innovation

• Recoloring via single vertex color changes
• Lists require size degree plus one
• Phase transition in Glauber dynamics