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2-packing 3-block configuration additive permutations adjacent assume automorphism Auxiliary square binomial coefficients bipartite graph blocks cardinality chromatic index clique codeword coloring column Combinatorial complete graph component conjecture connected graph consider construction contradiction Corollary counterexamples cycle of length cyclic defined degree denote Department of Mathematics digraph disjoint dominating set DTS(v edge edge coloring elements equation exactly exists Figure finite follows formulae found C(l G E(G graph G graph of order Graph Theory grid Hamiltonian path Hence homomorphism induced subgraph integer isomorphic labeled least Lemma Let G matrix maximal minor-minimal niche graph NP-complete obtained orthogonal pair parameters partition Petersen graph points polynomial positive integer problem proof of Theorem Proposition prove result seed graph self-dual code sequence spanning Stirling numbers subgraph of G subset Suppose symplectic traceable scenic graph triangle-free triangle-free graph triple systems vector vertex vertex set