Number of independent transversals in multipartite graphs
Yantao Tang, Yi Zhao
TL;DR
The paper addresses the number and existence of independent transversals in $r$-partite graphs under degree constraints. It proves that for even $r$, the minimum number of independent transversals $f_r(n,t)$ is asymptotically proportional to $t n^{r-1}$ when $t=o(n)$, thereby confirming the Haxell–Szabó conjecture and answering Erdős’ 1972 question in this regime. It also establishes a blowup result: for any even $r$ and $s\ge2$, there exist constants so that if $\Delta(G)\le \frac{r}{2r-2}n- c_{r,s}n^{1-1/s}$, then $G$ contains an IT$(s)$, with the bounds shown to be tight up to constants under plausible Zarankiewicz conjectures. The approach combines the Graph Removal Lemma, HS2006 structure theory for graphs without IT, hypergraph Turán-type results (K_r^r(s)), and Zarankiewicz bounds to derive both lower and upper bounds and to construct near-optimal extremal graphs. These results advance the understanding of how independence, transversals, and blowups interact in multipartite graphs and connect to longstanding extremal problems.
Abstract
An independent transversal in a multipartite graph is an independent set that intersects each part in exactly one vertex. We show that for every even integer $r\ge 2$, there exist $c_r>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)-t$, where $t=o(n)$, contains at least $c_r t n^{r-1}$ independent transversals. This is best possible up to the value of $c_r$. Our result confirms a conjecture of Haxell and Szabó from 2006 and partially answers a question raised by Erdős in 1972 and studied by Bollobás, Erdős and Szemerédi in 1975. We also show that, given any integer $s\ge 2$ and even integer $r\ge 2$, there exist $c_{r,s}>0$ and $n_0$ such that every $r$-partite graph with parts of size $n\ge n_0$ and maximum degree at most $rn/(2r-2)- c_{r, s} n^{1-1/s}$ contains an independent set with exactly $s$ vertices in each part. This is best possible up to the value of $c_{r, s}$ if a widely believed conjecture for the Zarankiewicz number holds. Our result partially answers a question raised by Di Braccio and Illingworth recently.