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1-factor adjacency matrices algorithm for constructing array assigned associativity basic unit bichromatic binary trees block C V(G Cayley table columns combinatorial Computer Science connected component containing corresponding Cutting Stock Problem defined denote edge color edge of color edge-coloring elements entries example expected number Figure finite first-fit given groupoid Hamiltonian cycle Hibbard's sequence hole initial run input stack insertion sorting integer intersection of three intersection problem inverse k-parity problem Kleitman Knuth Lemma loops lower bound merge step monochromatic triangles multiplication nodes number of operations optimal pair paper parity set partial ordering partition matroids passengers permutation polynomial-bounded algorithm proof Ramsey Numbers Ramsey's Theorem random Boolean matrix rank group regular coloring result rithm semigroup Shellsort spanning tree subgraph subsets symmetric sum-free sets tape template Theorem tion transitive closure triangles of color trichromatic triangles triples upper bound valence values vertex vertices weighted path length Whitehead