Finite-Size Scaling in q-Coloring Problems

We numerically investigate q-coloring (q-COL) problems in random networks in terms of a stochastic-local-search (SLS) algorithm. Random q-COL problems involve finding solutions where all nodes consist of different colors from their neighbors' colors, among q colors. For a fixed number of colors, say q = 3 or larger, various phase transitions have been reported as the average degree of links (the number density of constraints), c = , increases. This is because the set of solutions undergoes several types of phase transitions similar to those observed in the mean-field theory of spin glasses at zero temperature. Eventually, a dynamic coloring threshold is found to exist, above which no more solutions exist. Using the finite-size scaling (FSS) technique for nonequilibrium absorbing phase transitions, we analyze critical behaviors in the dynamic phase transition of q-COL problems by using the SLS algorithm, where we test both $Erd{\ddot{o}}s-R{\acute{e}}nyi$ and regular random networks. Finally, we discuss the extended FSS in q-COL problems compared to random k-satisfiability ones.