Fractional triangle decompositions of dense 3-partite graphs

TL;DR

This paper determines the minimum degree threshold needed for dense 3-partite graphs to have a fractional triangle decomposition, advancing understanding of graph decompositions and their applications to Latin squares.

Contribution

It provides a precise threshold for fractional triangle decompositions in dense 3-partite graphs, extending previous results and implications for exact decompositions.

Findings

Established a minimum degree threshold for fractional triangle decompositions.

Connected fractional decompositions to exact decompositions and Latin square completion.

Extended the theory to include some additional graph classes or conditions.

Abstract

We compute a minimum degree threshold sufficient for 3-partite graphs to admit a fractional triangle decomposition. Together with recent work of Barber, K\"uhn, Lo, Osthus and Taylor, this leads to bounds for exact decompositions and in particular the completion problem for sparse partial latin squares. Some extensions are considered as well.