430 research outputs found
Some Implications on Amorphic Association Schemes
AMS classifications: 05E30, 05B20;amorphic association scheme;strongly regular graph;(negative) Latin square type;cyclotomic association scheme;strongly regular decomposition
Some Implications on Amorphic Association Schemes
AMS classifications: 05E30, 05B20;
Hypergraph matchings and designs
We survey some aspects of the perfect matching problem in hypergraphs, with
particular emphasis on structural characterisation of the existence problem in
dense hypergraphs and the existence of designs.Comment: 19 pages, for the 2018 IC
Linearly many rainbow trees in properly edge-coloured complete graphs
A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different
colours. The study of rainbow decompositions has a long history, going back to the work of
Euler on Latin squares. In this paper we discuss three problems about decomposing complete
graphs into rainbow trees: the Brualdi-Hollingsworth Conjecture, Constantineās Conjecture, and
the Kaneko-Kano-Suzuki Conjecture. We show that in every proper edge-colouring of Kn there
are 10ā6n edge-disjoint spanning isomorphic rainbow trees. This simultaneously improves the
best known bounds on all these conjectures. Using our method we also show that every properly
(n ā 1)-edge-coloured Kn has n/9 ā 6 edge-disjoint rainbow trees, giving further improvement on
the Brualdi-Hollingsworth Conjectur
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