5 research outputs found
Classification of maximum hittings by large families
For integers and , where is sufficiently large, and for every set
we determine the maximal left-compressed intersecting
families which achieve maximum hitting
with (i.e. have the most members which intersect ). This answers a
question of Barber, who extended previous results by Borg to characterise those
sets for which maximum hitting is achieved by the star.Comment: v2: minor corrections in response to reviewer comments. To appear in
Graphs and Combinatoric
A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs
The study of asymptotic minimum degree thresholds that force matchings and
tilings in hypergraphs is a lively area of research in combinatorics. A key
breakthrough in this area was a result of H\`{a}n, Person and Schacht who
proved that the asymptotic minimum vertex degree threshold for a perfect
matching in an -vertex -graph is
. In this paper we improve on this
result, giving a family of degree sequence results, all of which imply the
result of H\`{a}n, Person and Schacht, and additionally allow one third of the
vertices to have degree below this threshold.
Furthermore, we show that this result is, in some sense, tight.Comment: 21 page
Universality for transversal Hamilton cycles
Let be a graph collection on a common
vertex set of size such that for every
. We show that contains every Hamilton cycle pattern.
That is, for every map there is a Hamilton cycle whose
-th edge lies in .Comment: 18 page
A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs
The study of asymptotic minimum degree thresholds that force matchings and tilings in hypergraphs is a lively area of research in combinatorics. A key breakthrough in this area was a result of HΓ n, Person, and Schacht [SIAM J. Disc. Math., 23 (2009), pp. 732--748] who proved that the asymptotic minimum vertex degree threshold for a perfect matching in an -vertex -graph is . In this paper, we improve on this result, giving a family of degree sequence results, all of which imply the result of HΓ n, Person and Schacht and additionally allow one-third of the vertices to have degree below this threshold. Furthermore, we show that this result is, in some sense, tight