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ExaGraph Co-Design Center of the Department of Energy Exascale Computing Project (ECP) uses combinatorial kernels, with key examples being smart power grid, computational biology, computational chemistry, wind energy, and national security. Combinatorial (graph) kernels that play crucial enabling roles in the chosen application areas will be co-designed: graph traversals; graph matching; graph coloring; graph clustering, parallel branch-and-bound, and graph partitioning.