# Sublinear Algorithms for (Delta + 1) Vertex Coloring

Sepehr Assadi (University of Pennsylvania)

Any graph with maximum degree Delta admits a proper vertex coloring with Delta+1 colors that can be found via a simple sequential greedy algorithm in linear time and space. But can one find such a coloring via a sublinear algorithm? In this talk, I present new algorithms that answer this question in the affirmative for several canonical classes of sublinear algorithms including graph streaming, sublinear time, and massively parallel computation (MPC) algorithms. At the core of these algorithms is a remarkably simple meta-algorithm for the (Delta+1) coloring problem: Sample O(log n) colors for each vertex uniformly at random from the Delta+1 colors and then find a proper coloring of the graph using the sampled colors; our main structural result states that the sampled set of colors with high probability contains a proper coloring of the input graph.