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1-factors acyclic coloring adjacent vertices assign the color bichromatic cycle bipartite graph blocks Combinatorics complete graph Computer conjecture connected graph construction contains contingency sets contradiction Corollary cyclic difference sets defined degFl denoted designs Discrete Math domination subdivision number domino magic edge coloring edge labelled edge-graceful edges of length elements Euler Medal example exists Golomb rulers graceful labelling graph G graph of order Graph Theory Hamilton cycles Hence incident Institute of Combinatorics irredundant set Kirkman Medal latin squares least Lemma Let G magic rectangle magic square maximum minimum mutually orthogonal nodes number of edges obtained pair parallel class partition path planar graph private neighbor problem Proof Proposition segment sequence Steiner triple systems Subcase subgraph Sudoku Suppose Theorem total dominating set tournament triple systems University vertex vertex of degree vertex set Walecki weakly connected domination Xd(G Xir(G