An Asymptotically Sharp Bound on the Maximum Number of Independent Transversals

TL;DR

This paper establishes an asymptotically sharp upper bound on the maximum number of independent transversals in a multipartite graph based on the edge densities between its parts.

Contribution

It provides a new theoretical bound on independent transversals in multipartite graphs, extending understanding of their combinatorial limits.

Findings

Derived an asymptotically sharp upper bound for independent transversals.

Connected edge densities to the maximum number of independent transversals.

Enhanced theoretical understanding of multipartite graph structures.

Abstract

Let $G$ be a multipartite graph with partition $V_1, V_2,\ldots, V_k$ of $V(G)$. Let $d_{i,j}$ denote the edge density of the pair $(V_i, V_j)$. An independent transversal is an independent set of $G$ with exactly one vertex in each $V_i$. In this paper, we prove an asymptotically sharp upper bound on the maximum number of independent transversals given the $d_{i,j}$'s.