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1-tough 2-coloring 2-competition additive permutations adjacency matrix algorithm antichains assigned association scheme B-array back-edge bijection cardinality chessboard graphs clique Combinatorial complete graphs components computing consider construct contradiction control dependence crease cross-edge cut vertex cycle equivalent defined degree sequence denote digraph directed graph disjoint dominating set edges colored edges incident edges of G eigenvalues element equations equivalent in G exactly exists Figure flat origami functional program given graph G green vertex Hamiltonian circuits Hence hypergraph implies independent set integer isomorphic Lemma Let G linear marked vertex mixed domination node nullspace number of edges number of vertices NW(v obtained optimum coloring orientation of G origami P-complete p-conforming parameters partition path planar graphs problem proof of Theorem rank regular graph result rook's graph set of vertices subsets Suppose t-tough graph Theorem 1.3 tournament tree tree-path unilateral orientation unique upper bound vertex set vertices of degree Z-cyclic