On factors of independent transversals in k-partite graphs

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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 n0 such that every [k, n, 1]-graph has a factor of independent transversals assuming n ≥ n0 . Several known conjectures imply that for k ≥ 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) ≤ 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) ≤ 1.78k for all k sufficiently large, answering a question of MacKeigan.

Original languageEnglish
Article numberP4.23
JournalElectronic Journal of Combinatorics
Volume28
Issue number4
DOIs
StatePublished - 2021

Bibliographical note

Funding Information:
∗Supported by Israel Science Foundation (grant No. 1082/16).

Publisher Copyright:
© The author.

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics
  • Applied Mathematics

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