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3-colouring adjacent algorithm approximation bipartite co-partition co-solution codes colour class Combinatorial complete graph component Computer condition conjecture connected consider construction contains contradiction countable cross-free cubic graph defining set degree deleting denote the number edge-disjoint edges of G elements embedding example exists finite football pool given graph G Graph Theory Hamiltonian Hamiltonian graphs hash family Hence inequality infinite internally disjoint isomorphic latin square least Lemma Let G lower bound Mathematics matrix matroid minimum Moore Graphs N-free posets Nash-Williams node number of edges obtained orientation pair panconnected paper partition path planar graph posets positive integers proof of Theorem prove result satisfies solution spanning Steiner triple systems subgraph of G submodular flow subset supermodular Suppose Theorem 1.2 toroidal graphs total chromatic total chromatic number transversal tree triple UE3CC unique upper bound V-order vectors vertex vertex set weakly complete