Independent transversals in r-partite graphs

Let G(r, n) denote the set of all r-partite graphs consisting of n vertices in each partite class. An independent transversal of G ∈ G(r, n) is an independent set consisting of exactly one vertex from each vertex class. Let ∆(r, n) be the maximal integer such that every G ∈ G(r, n) with maximal degree less than ∆(r, n) contains an independent transversal. Let Cr = limn→∞ ∆(r,n) n . We establish the following upper and lower bounds on Cr, provided r > 2: 2blog rc−1 2blog rc − 1 ≥ Cr ≥ max{ 1 2e , 1 2dlog(r/3)e , 1 3 · 2dlog re−3 }. For all r > 3, both upper and lower bounds improve upon previously known bounds of Bollobás, Erdös and Szemerédi. In particular, we obtain that C4 = 2/3, and that limr→∞ Cr ≥ 1/(2e), where the last bound is a consequence of a lemma of Alon and Spencer. This solves two open problems of Bollobás, Erdös and Szemerédi.