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Amer Barbara Beineke bicolored graph cell chromatic number classes colors complete bipartite graph complete graph component game configuration connected graph Corollary counting polynomial cycle index cycles of length defined definition degree denote digraphs disjoint disjunctive compound entries enumeration Erdos Euler's polyhedron formula finite follows force a win formula four-color conjecture graph contains graph G graph G(n graph theory Gu(Vi H(Vs hamiltonian Harary Hence homeomorphic Hu(Vi impartial game induced infinite graph isomorphic labeled points last-player-winning least Lemma lines of G Math nonnegative integer number of points number of trees obtained ordered pairs pair group path joining Paul Erdos permutation group planar graph play player points of G Polya's theorem problems in graph proof Proposition proved R(Vx regular graph remoteness function result rooted trees sequence shows strategy subgraph of G subsets suppose terminal distance terminal position thickness triangle well-quasi-ordered