Partial independent transversals in multipartite graphs

TL;DR

This paper extends classical results on independent transversals in multipartite graphs by establishing new tight bounds for the existence of partial independent transversals under various conditions, generalizing previous work.

Contribution

The paper introduces new tight bounds for the existence of partial independent transversals in multipartite graphs, extending and generalizing classical results by Haxell and others.

Findings

Established bounds for $(r-d)$-independent transversals depending on part size and degree.

Extended classical results to broader parameter ranges with tight bounds.

Answered an open question regarding 5-IT in 6-partite graphs.

Abstract

Given integers $r>d\ge 0$ and an $r$-partite graph, an independent $(r-d)$-transversal or $(r-d)$-IT is an independent set of size $r-d$ that intersects each part in at most one vertex. We show that every $r$-partite graph with maximum degree $\Delta$ and parts of size $n$ contains an $(r-d)$-IT if $n> 2\Delta (1-\frac{1}{q})$, provided $q= \lfloor \frac{r}{d+1}\rfloor\ge \frac{4r}{4d+5}$. This is tight when $q$ is even and extends a classical result of Haxell in the $d=0$ case. When $q= \lfloor \frac{r}{d+1} \rfloor\ge \frac{6r+6d+7}{6d+7}$ is odd, we show that $n> 2\Delta(1-\frac{1}{q-1})$ guarantees an $(r-d)$-IT in any $r$-partite graph. This is also tight and extends a result of Haxell and Szab\'o in the $d=0$ case. In addition, we show that $n> 5\Delta/4$ guarantees a $5$-IT in any $6$-partite graph and this bound is tight, answering a question of Lo, Treglown and Zhao.