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3RD S-E CONF adjacent algebra algorithm alternating edges approximation assume average number block design calculated cell chromatic polynomial closed braids coefficients COMBINATORICS complete block complete subgraph congruent number conjecture connected contains Corollary crossing number cycle defined denote determine Diophantine equation elements endpoints equation exist factor Figure finite Florida Atlantic University given graph G GRAPH THEORY hashing function Hence hypohamiltonian graphs hypotraceable graph integer irreducible KNOT label Latin squares lattice Lemma linear major rotation Math matrix matroid method minimal modulo nodes number of edges number of probes obtained occur odd chords P,Q)-homomorphism P,Q)-system pair parameters partial cut partition planar graph polynomial prime problem PROC Proof quadratic secondary flat solution solve spanning path spline square free square of side subgraph of G subsets symbol tree edge type S Latin uniform edges University of Manitoba University of Waterloo values vector vertex vertices of valence