We analyze uniformly random suitable $k$ color schemes for sparse graphs with maximal degree $\Delta$ in the regime $\Delta < k\ln k$. This regime corresponds to the lower crushing threshold for random graph coloring and is a paradigmatic example of crushing threshold for random constraint satisfaction problems. We prove various results on the solution space geometry of the coloring of fixed graphs, generalize his work of Achlioptas, Coja-Oghlan, and Molloy on random his graphs, and justify the performance of probabilistic local search algorithms in this regime. become Our central proof relies only on basic techniques, i.e., the method of first instants and quantitative induction, but the result of list coloring by Vu, and more recently Davies, Kang, P., and Sereni Reinforcing the state-of-the-art boundary from Ramsey theory in the context of sparse graphs. In addition, we obtain a near-strict lower bound on the coloration number, also known as the Potts model partition function, which affects efficient approximate counting.
