Rainbow Numbers for Graphs Containing Small Cycles

Abstract

For a given graph $$H$$H and $$n \ge 1,$$n?1, let $$f(n, H)$$f(n,H) denote the maximum number $$m$$m for which it is possible to colour the edges of the complete graph $$K_n$$Kn with $$m$$m colours in such a way that each subgraph $$H$$H in $$K_n$$Kn has at least two edges of the same colour. Equivalently, any edge-colouring of $$K_n$$Kn with at least $$rb(n,H)=f(n,H) + 1$$rb(n,H)=f(n,H)+1 colours contains a rainbow copy of $$H.$$H. The numbers $$f(n,H)$$f(n,H) and $$rb(K_n,H)$$rb(Kn,H) are called anti-ramsey numbers and rainbow numbers, respectively. In this paper we will classify the rainbow number for a given graph $$H$$H with respect to its cyclomatic number. Let $$H$$H be a graph of order $$p \ge 4$$p?4 and cyclomatic number $$v(H) \ge 2.$$v(H)?2. Then $$rb(K_n, H)$$rb(Kn,H) cannot be bounded from above by a function which is linear in $$n.$$n. If $$H$$H has cyclomatic number $$v(H) = 1,$$v(H)=1, then $$rb(K_n, H)$$rb(Kn,H) is linear in $$n.$$n. Moreover, we will compute all rainbow numbers for the bull $$B,$$B, which is the unique graph with $$5$$5 vertices and degree sequence $$(1,1,2,3,3)$$(1,1,2,3,3).