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adjacent algorithm Anne Penfold assigned bipartite graphs block design caterpillar cell chromatic index chromatic spectrum combinatorial design Combinatorics complete graph Computer conference conjecture connected graph construction contains copy of G Corollary critical independent set critical weighted cycle cyclic Darryn decompositions defined degree denote Department of Mathematics diagonal diamG digraphs Discrete Mathematics disjoint dispemorphism dominating set domination number edges elements Euler Medal exists Figure finite graph G graph of order Graph Theory Harary Hence induced subgraph Institute of Combinatorics Iran irregular chromatic number isomorphic Kirkman Medals Kq+i labelled Latin squares Lemma matrix maximum independent set Multidecomposition Note pair partite path positive integer prime problem projective plane Proof R.G. Stanton r)-regular Ralph Faudree representation resolvable spanning trees spinal nodes Steiner subgraph subset subtree Sudoku Sudoku squares Triple Systems TS(v vertex vertices of G weighted independent set Xir(Cn Xir(G