Rainbow Numbers for Graphs Containing Small Cycles
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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 \((1,1,2,3,3)\).