On an Anti-Ramsey Property of Ramanujan Graphs

If G and H are graphs, we write G → H (respectively, G → TH) if for any proper edge-colouring γ of G there is a subgraph H ′ ⊂ G of G isomorphic to H (respectively, isomorphic to a subdivision of H) such that γ is injective on E(H ′). Let us write C for the cycle of length l. Spencer (cf. Erdős [10]) asked whether for any g ≥ 3 there is a graph G = Gg such that (i) G has girth g(G) at least g, and (ii) G → TC. Recently, Rödl and Tuza [22] answered this question in the affirmative by proving, using non-constructive methods, a result that implies that for any t ≥ 1 there is a graph G = Gt of girth t + 2 such that G → C . In particular, condition (ii) may be strengthened to (iii) G → C for some l = l(G). For G = Gt above l = l(G) = 2t + 2 = 2g(G) − 2. Here, we show that suitable Ramanujan graphs constructed by Lubotzky, Phillips, and Sarnak [18] are explicit examples of graphs G = Gg satisfying (i) and (iii) above. For such graphs l = l(G) in (iii) may be taken to be roughly equal to (3/2)g(G), thus considerably improving the value 2g(G)−2 given in the result of Rödl and Tuza. It is not known whether there are graphs G of arbitrarily large girth for which (iii) holds with l = l(G) = g(G).