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A gem is a graph that consists of a path on four vertices plus a vertex adjacent to all four vertices of the path. A co‐gem is the complement of a gem. We prove that every (gem, co‐gem)‐free graph G satisfies the inequality (a special case of a conjecture of Gyárfás) and the inequality (a special case of a conjecture of Reed). Moreover, we give an ‐time algorithm...
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