Asymptotic behaviour of the complexity of coloring sparse random graphs∗

The behaviour of a backtrack algorithm for graph coloring is well understood for large random graphs with constant edge density. However, sparse graphs, in which the edge density decreases with increasing graph size, are more common in practice. Therefore, in this paper we analyze the expected runtime of a usual backtrack search to color such random graphs, when the size of the graph tends to infinity. Contrary to the case of constant edge density, where the expected runtime is known to be O(1), here we prove that the expected runtime tends to infinity in this case. We also examine when the expected runtime grows polynomially or exponentially, depending on the edge density function. Besides, we also investigate the asymptotic behaviour of the expected number of solutions in this model.