Quasi-transversals in Latin squares and k-domination in graphs

Document Type : Original Article

Authors

1 Associate Professor, Faculty of Mathematics, Mohaghegh Ardabili University, Ardabil, Iran

2 Ph.D., Faculty of Mathematics, Hokkaido University, Sapporo, Japan

Abstract

In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem involving Combinatorics is to find maximum number of disjoint transversals in a Latin square. So finding some family of decomposible Latin squares into disjoint transversals is our next aim. In the next section, we give an equivalent statement of a conjecture which has been attributed to Brualdi, Stein and Ryser by the concept of quasi-transversal. Finally, we prove the truth of the Rodney's conjecture for a family of graphs.

Keywords

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References

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History

  • Receive Date: 27 October 2021
  • Revise Date: 14 December 2021
  • Accept Date: 09 August 2022
  • Publish Date: 22 December 2022

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