Topological Minors in Graphs of Large Girth

We prove that every graph of minimum degree at least r and girth at least 186 contains a subdivision of Krþ1 and that for r5435 a girth of at least 15 suffices. This implies that the conjecture of Haj ! os that every graph of chromatic number at least r contains a subdivision of Kr (which is false in general) is true for graphs of girth at least 186 (or 15 if r5436). More generally, we show that for every graph H of maximum degree DðH Þ52; every graph G of minimum degree at least maxfDðH Þ; 3g and girth at least 166 log jH j log DðH Þ contains a subdivision of H : This bound on the girth of G is best possible up to the value of the constant and improves a result of Mader, who gave a bound linear in jH j: # 2002 Elsevier Science (USA)