On Factors of Independent Transversals in $k$-Partite Graphs

A $[k,n,1]$-graph is a $k$-partite graph with parts of order $n$ such that the bipartite induced by any pair matching. An independent transversal in an set intersects each part single vertex. factor transversals pairwise-disjoint transversals. Let $f(k)$ be smallest integer $n_0$ every has assuming $n \ge n_0$. Several known conjectures imply for $k 2$, $f(k)=k$ if $k$ even and $f(k)=k+1$ odd. While simple greedy algorithm based on iterating Hall's Theorem shows $f(k) \le 2k-2$, no better bound fact, there are instances showing $2k-2$ tight algorithm. Here we significantly improve upon prove 1.78k$ all sufficiently large, answering question MacKeigan.