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The purpose of this note is to present a polynomial-time algorithm which, given an arbitrary graph G as its input, finds either a proper 3-coloring of G or an odd-K 4 that is a subgraph of G in time O(mn), where m and n stand for the number of edges and the number of vertices of G, respectively.
Grillet established conditions on a partially ordered set under which each maximal antichain meets each maximal chain. Berge pointed out that Grillet's theorem can be stated in terms of graphs, made a conjecture that strengthens it, and asked a related question. We exhibit a counterexample to the conjecture and answer the question; then we prove four theorems that generalize Grillet's theorem in...