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The Product of Three Cycles
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adjacent an+2 an+i Assign the pair assume block cartesian product cells choose color appears coloring of G column complete bipartite graph complete graph completes the proof component coloring connected acyclic partition contradict p'(G contradicting the hypothesis corresponding cross-edges cubic graphs cut-vertex cycle in G decomposition denote disjoint disjoint sets edge-coloring edges of G endpoints entry follows form a partition G with less gives p'(G graph color graph G graph of order hamiltonian path hypothesis on G indicates the coloring integer Kn(u Kronecker product Lemma less than n/2 Let G linking paths minimum number multiple edges n/2 sets number of hamiltonian odd endblock ordered pair orthogonal pair of colors pairs to contradict partition for G planes with copies r-factors red blue s(Kn similarly skew chromatic index skew edge coloring skew Room square Subcase Theorem three hamiltonian cycles Tnen unordered pairs upper bound vertex coloring vertices of G