Rainbow Numbers for Graphs Containing Small Cycles

Abstract

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