Abstract:We consider the structure of K r -free graphs with large minimum degree, and show that such graphs with minimum degree >(2r−5)n / (2r−3) are homomorphic to the join K r−3 ∨H, where H is a triangle-free graph. In particular this allows us to generalize results from triangle-free graphs and show that K r -free graphs with such a minimum degree have chromatic number at most r+1. We also consider the minimum-degree thresholds for related properties. ᭧

A set S of vertices in a graph G is an independent dominating set of G if S is an independent set and every vertex not in S is adjacent to a vertex in S. In this paper, we consider questions about independent domination in regular graphs.

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