The dynamics of proving uncolourability of large random graphs: I. Symmetric colouring heuristic

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, ω(c) = (In T)/N, is quantitatively predicted, in agreement with simulations.