Rainbow Numbers for Graphs Containing Small Cycles

For a given graph H and n ≥ 1, let f(n,H) denote the maximum number m for which it is possible to colour the edges of the complete graph Kn with m colours in such a way that each subgraph H in Kn has at least two edges of the same 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 talk we will classify the rainbow number for a given graph H with respect to its cyclomatic 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 linear in n. If H has cyclomatic number v(H) = 1, then rb(Kn, H) is linear in n. We will compute all rainbow numbers for the bull B, which is the unique graph with 5 vertices and degree sequence (1, 1, 2, 3, 3). Furthermore, we will compute some rainbow numbers for the diamond D = K4 − e, for K2,3, and for the house H = P5.