32 research outputs found
Rainbow matchings in properly-coloured multigraphs
Aharoni and Berger conjectured that in any bipartite multigraph that is
properly edge-coloured by colours with at least edges of each
colour there must be a matching that uses each colour exactly once. In this
paper we consider the same question without the bipartiteness assumption. We
show that in any multigraph with edge multiplicities that is properly
edge-coloured by colours with at least edges of each colour
there must be a matching of size that uses each colour at most once.Comment: 7 page
Sparse halves in dense triangle-free graphs
Erd\H{o}s conjectured that every triangle-free graph on vertices
contains a set of vertices that spans at most
edges. Krivelevich proved the conjecture for graphs with minimum degree at
least . Keevash and Sudakov improved this result to graphs with
average degree at least . We strengthen these results by showing
that the conjecture holds for graphs with minimum degree at least
and for graphs with average degree at least for some absolute . Moreover, we show that the
conjecture is true for graphs which are close to the Petersen graph in edit
distance.Comment: 23 page
On the number of symbols that forces a transversal
Akbari and Alipour conjectured that any Latin array of order with at
least symbols contains a transversal. We confirm this conjecture for
large , and moreover, we show that symbols suffice.Comment: 6 pages, 1 figur
On The Random Tur\'an number of linear cycles
Given two -uniform hypergraphs and the Tur\'an number is the maximum number of edges in an -free subgraph of . We study the
typical value of when , the Erd\H{o}s-R\'enyi
random -uniform hypergraph, and , the -uniform linear
cycle of length . The case of graphs () is a longstanding open
problem that has been investigated by many researchers. We determine the order
of magnitude of for all
and all up to polylogarithmic factors for all values of
.
Our proof is based on the container method and uses a balanced
supersaturation result for linear even cycles which improves upon previous such
results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan.Comment: 16 pages. arXiv admin note: substantial text overlap with
arXiv:2007.1032