Cycles of even lengths modulo k

Thomassen [J Graph Theory 7 (1983), 261–271] conjectured that for all positive integers k and m , every graph of minimum degree at least k +1 contains a cycle of length congruent to 2 m modulo k . We prove that this is true for k ⩾2 if the minimum degree is at least 2 k −1, which improves the previously known bound of 3 k −2. We also show that Thomassen's conjecture is true for m = 2. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 246–252, 2010