Independent transversals in r-partite graphs

Abstract

Let $\text{G}\text{(r,n)}$ denote the set of all r-partite graphs consisting of n vertices in each partite class. An independent transversal of $\text{G ∈}\text{G}\text{(r,n)}$ is an independent set consisting of exactly one vertex from each vertex class. Let Δ(r,n) be the maximal integer such that every $\text{G ∈}\text{G}\text{(r,n)}$ with maximal degree less than Δ(r,n) contains an independent transversal. Let C_r = lim_n→∞ Δ(r,n)/n. We establish the following upper and lower bounds on C_r, provided r > 2:

$$\text{2⌊log r⌋−1}\text{2⌊log r⌋−1}\text{⩾C}{}_{\mathrm{r}}\text{⩾max{}\text{1}\text{2e}\text{,}\text{1}\text{2⌈log(}\text{r}\text{3}\text{)⌉}\text{,}\text{1}\text{3 · 2⌈log r⌉−3}\text{}}$$

. For all r > 3, both upper and lower bounds improve upon previously known bounds of Bollobás, Erdős and Szemerédi. In particular, we obtain that $\text{C}{}_4\text{=}\text{2}\text{3}$, and that lim_r→∞ C_r ⩾ 1/(2e), where the last bound is a consequence of a lemma of Alon and Spencer. This solves two open problems of Bollobás, Erdős and Szemerédi.