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adjacent algorithm assignment automorphism bipartite graph Cayley graph chordal graphs chromatic classes clique cocritical coloring Combinatorial complete graph computed conjecture connected graph consider construction Corollary critically n-connected cubic graphs cycle cyclic defined degree deleting denote diameter digraph Discrete Math disjoint dominating set eigenvalue embedding example exists family of graphs Figure finite function G contains given graph G graph of order Graph Theory hamiltonian Harary Hence homomorphism hypercube hypergraph implies intersection graph interval graphs irregularity strength isomorphic it-tournaments labeled least Lemma Let G lower bound mapping matching matrix maximal independent sets maximum minimum multigraph multiplicity MXST(G nodes nonadjacent vertices nonorientable number of vertices obtained odd components orthomorphism pair partition path Petersen graph planar graph plane points polynomial problem Proof regular graphs rooted satisfies sequence snarks spanning tree spanning unicyclic subgraphs subset Suppose symmetric T-colorings Theorem upper bound vertex vertex set walk