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 \((1,1,2,3,3)\).
KeywordsRainbow colouring Rainbow number Anti-ramsey number Turán number
We thank Jana Neupauerová for some stimulating discussions on the computation of the rainbow numbers for the bull and an anonymous referee for some valuable comments.
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