Polynomial iterative algorithms for coloring and analyzing random graphs

We study the graph coloring problem over random graphs of finite average connectivity c. Given a number q of available colors, we find that graphs with low connectivity admit almost always a proper coloring whereas graphs with high connectivity are uncolorable. Depending on q, we find with a one-step replica-symmetry breaking approximation the precise value of the critical average connectivity $c_q.$ Moreover, we show that below $c_q$ there exists a clustering phase $c\in \left[c_d{,c}_q\right]$ in which ground states spontaneously divide into an exponential number of clusters. Furthermore, we extended our considerations to the case of single instances showing consistent results. This leads us to propose a different algorithm that is able to color in polynomial time random graphs in the hard but colorable region, i.e., when $c\in \left[c_d{,c}_q\right].$