On the Anti-Ramsey Property of Ramanujan Graphs

If G and H are graphs, we write G→ H (respectively, G→ TH) if for any proper edge‐coloring 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 Cl for the cycle of length l. Spencer (cf. Erdos 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→ TC3. Recently, Rodl and Tuza [22] answered this question in the affirmative by proving, using nonconstructive methods, a result that implies that, for any t ⩾ 1, there is a graph G = Gt of girth t + 2 such that G→ C2t+2. In particular, condition (ii) may be strengthened to (iii) G→ C𝓁 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 Rodl 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). © 1995 Wiley Periodicals, Inc.