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For a given graph H and n ? 1; let f(n;H) denote the maximum number m forwhich it is possible to colour the edges of the complete graph Kn with mcolours in such a way that each subgraph H in Kn has at least two edges of thesame colour. Equivalently, any edge-colouring of Kn with at least rb(n;H) =f(n;H)+1 colours contains a rainbow copy of H: The numbers f(n;H) and rb(Kn;H)are called anti-ramsey numbers and rainbow numbers, respectively. In this paperwe will classify the rainbow number for a given graph H with respect to itscyclomatic number. Let H be a graph of order p >= 4 and cyclomatic number v(H)>= 2: Then rb(Kn;H) cannot be bounded from above by a function which is linearin n: If H has cyclomatic number v(H) = 1; then rb(Kn;H) is linear in n: Wewill compute all rainbow numbers for the bull B; which is the unique graph with5 vertices and degree sequence (1; 1; 2; 3; 3): Furthermore, we will computesome rainbow numbers for the diamond D = K_4 - e; for K_2;3; and for the houseH (complement of P_5).

Rainbow numbers for graphs with cyclomatic number at most two