The dynamics of proving uncolourability of large random graphs I. Symmetric Colouring Heuristic

TL;DR

This paper analyzes the average running time of a backtracking algorithm for proving uncolourability in large sparse random graphs, using a novel surface growth model to predict its behavior.

Contribution

It introduces a new analytical approach by mapping the backtracking process onto a surface growth problem to predict algorithm complexity.

Findings

Predicted growth exponent matches simulation results.

Average running time depends on graph density and size.

Provides insights into the complexity of graph uncolourability proofs.

Abstract

We study the dynamics of a backtracking procedure capable of proving uncolourability of graphs, and calculate its average running time T for sparse random graphs, as a function of the average degree c and the number of vertices N. The analysis is carried out by mapping the history of the search process onto an out-of-equilibrium (multi-dimensional) surface growth problem. The growth exponent of the average running time is quantitatively predicted, in agreement with simulations.