Combinatorial algorithms in general and graph algorithms in particular play a critical enabling role in numerous scientific applications. The irregular memory access nature of these algorithms makes them one of the hardest algorithmic kernels to implement on parallel systems. We therefore propose to develop methods and techniques for efficient implementation of key combinatorial (graph) algorithms chosen from a set of exascale applications. ExaGraph is one of the codesign centers funded by the Exascale Computing Project (ECP) that focuses on key combinatorial and graph algorithms that are motivated from several exascale applications, and designs and develops scalable implementations for these algorithms.