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

Abstract

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

Packing list‐colorings

“…In this section we give the argument for Theorem 9, which also yields Theorem 3. This greedy approach with Hall's theorem is given by MacKeigan [31] and discussed by Yuster [42] as a way of proving that $$$ {\chi}_c^{\star}\left({K}_n\right)\leqslant 2\left(n-1\right) $$$. Although Yuster proved that the leading constant ‘2’ there is not tight, we show with Proposition 24 that when the method is refined and stated in terms of degeneracy it is indeed tight.…”

“…As a corollary of [10,Thr. 4] and a classical construction (compare [46]), χ ‹ c pK 5 q " 6. Assume G is not K 5 but contains K 5 as a subgraph.…”

“…In terms of the number of vertices, the problem of correspondence packing has recently been highlighted (in different terminology) by Yuster [42]. Casting it as a specialization of earlier conjectures related to the Hajnal–Szemerédi theorem, he stated the conjecture, rephrased in our terminology, that $$$ {\chi}_c^{\star}\left({K}_n\right)=n $$$ if $$$ n $$$ is even and $$$ {\chi}_c^{\star}\left({K}_n\right)=n+1 $$$ if $$$ n $$$ is odd.…”