Abstract:We study edge-labelings of the complete bidirected graph ↔ K n with functions from the set [d] = {1, . . . , d} to itself. We call a cycle in ↔ K n a fixed-point cycle if composing the labels of its edges results in a map that has a fixed point, and we say that a labeling is fixed-point-free if no fixed-point cycle exists. For a given d, we ask for the largest value of n, denoted R f (d), for which there exists a fixed-point-free labeling of ↔ K n . Determining R f (d) for all d > 0 is a natural Ramsey-type qu… Show more
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