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Extending the Affirmative Action Problem: mixing numbers and integrated colorings of graphs
Charles Burnette, Broden Caton, Olivia Coward, Julian Davis, Austin Teter
https://arxiv.org/abs/2506.07192

Extending the Affirmative Action Problem: mixing numbers and integrated colorings of graphs
Consider a graph whose vertices are colored in one of two colors, say black or white. A white vertex is called integrated if it has at least as many black neighbors as white neighbors, and similarly for a black vertex. The coloring as a whole is integrated if every vertex is integrated. A classic exercise in graph theory, known as the Affirmative Action Problem, is to prove that every finite simple graph admits an integrated coloring. The solution can be neatly summarized with the one-liner: "m…