178 research outputs found
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s on estimating the maximum number
of edges in a hypergraph when the (fractional) matching number is given, which
we are able to solve in some special cases using probabilistic techniques.
Based on these results, we obtain some general theorems on the minimum
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine the minimum vertex degree which
guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also
discuss an application to a problem of finding an optimal data allocation in a
distributed storage system
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
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
Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems
The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family {Mathematical expression} (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph Kn with t colors. Another area is to find the minimum number of monochromatic members of {Mathematical expression} that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of {Mathematical expression} in every edge coloring of Kn with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of Kn contains a (t - 1)-colored matching of size k provided that {Mathematical expression} © 2013 Springer Japan
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