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a b s t r a c tWe study the minimum number of weights assigned to the edges of a graph G with no component K 2 so that any two adjacent vertices have distinct sets of weights on their incident edges. The best possible upper bound on this parameter is proved.

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We show that every C 6 -free graph G has a C 4 -free, bipartite subgraph with at least 3e(G)/8 edges. Our proof is probabilistic and uses a theorem of Füredi, Naor and Verstraëte on C 6 -free graphs.

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