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4-valent graph achromatic number algorithm anti-automorphism Aut G automorphism group b-graceful Biggs Cayley map cell chain colouring heuristics Combinatorial Theory complete graph conjecture connected graph Corollary corresponding defined denote directed graph dual edge-colouring edges of G element embedded Euler trail Eulerian circuits Eulerian partition exact symmetric example faces of H Figure finite Frobenius group genus given graceful graph graceful numbering graph G graph imbedded graph satisfying Graph Theory induced subgraphs isomorphic junctions Kotzig labelled latin square Lemma Let H loops Math matroids non-square pair Paley graph Paley maps parameter q path crossings permutation group plane of order polynomial positive integers prime power problem projective plane proof of Theorem proved quasifield quasigroup region self-complementary graph self-dual imbedding spanning tree strongly regular subset symbol occurs symmetrical map Theorem Theorem 5.1 traffic transitive Tutte polynomial two-way infinite unique unital with parameter vertex voltage graph x)-latin rectangle x)-latin square