The effect of number of Hamiltonian paths on the complexity of a vertex-coloring problem

A generalization of Dobkin and Lipton´s element uniqueness problem is introduced: for any fixed undirected graph G on vertex set {v1, v2, ..., vn}, the problem is to determine, given n real numbers x1, x2, ..., xn, whether xi ≠ xj for every edge {vi, vj} in G. This problem is shown to have upper and lower bounds of Θ(nlogn) linear comparisons if G is any dense graph. The proof of the lower bound involves showing that any dense graph must contain a subgraph with many Hamiltonian paths, and demonstrating the relevance of these Hamiltonian paths to a geometric argument. In addition, we exhibit relatively sparse graphs for which the same lower bound holds, and relatively dense graphs for which a linear upper bound holds.