108 research outputs found
Rainbow Matchings and Hamilton Cycles in Random Graphs
Let be drawn uniformly from all -uniform, -partite
hypergraphs where each part of the partition is a disjoint copy of . We
let HP^{(\k)}_{n,m,k} be an edge colored version, where we color each edge
randomly from one of \k colors. We show that if \k=n and where
is sufficiently large then w.h.p. there is a rainbow colored perfect
matching. I.e. a perfect matching in which every edge has a different color. We
also show that if is even and where is sufficiently large
then w.h.p. there is a rainbow colored Hamilton cycle in . Here
denotes a random edge coloring of with colors.
When is odd, our proof requires m=\om(n\log n) for there to be a rainbow
Hamilton cycle.Comment: We replaced graphs by k-uniform hypergraph
Partitioning random graphs into monochromatic components
Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every -colored
complete graph can be partitioned into at most monochromatic components;
this is a strengthening of a conjecture of Lov\'asz (1975) in which the
components are only required to form a cover. An important partial result of
Haxell and Kohayakawa (1995) shows that a partition into monochromatic
components is possible for sufficiently large -colored complete graphs.
We start by extending Haxell and Kohayakawa's result to graphs with large
minimum degree, then we provide some partial analogs of their result for random
graphs. In particular, we show that if , then a.a.s. in every -coloring of there exists
a partition into two monochromatic components, and for if , then a.a.s. there exists an -coloring
of such that there does not exist a cover with a bounded number of
components. Finally, we consider a random graph version of a classic result of
Gy\'arf\'as (1977) about large monochromatic components in -colored complete
graphs. We show that if , then a.a.s. in every
-coloring of there exists a monochromatic component of order at
least .Comment: 27 pages, 2 figures. Appears in Electronic Journal of Combinatorics
Volume 24, Issue 1 (2017) Paper #P1.1
The Total Acquisition Number of Random Graphs
Let be a graph in which each vertex initially has weight 1. In each step,
the weight from a vertex can be moved to a neighbouring vertex ,
provided that the weight on is at least as large as the weight on . The
total acquisition number of , denoted by , is the minimum possible
size of the set of vertices with positive weight at the end of the process.
LeSaulnier, Prince, Wenger, West, and Worah asked for the minimum value of
such that with high probability, where
is a binomial random graph. We show that is a sharp threshold for this
property. We also show that almost all trees satisfy ,
confirming a conjecture of West.Comment: 18 pages, 1 figur
Large monochromatic components in expansive hypergraphs
A result of Gy\'arf\'as exactly determines the size of a largest
monochromatic component in an arbitrary -coloring of the complete
-uniform hypergraph when and . We prove a
result which says that if one replaces in Gy\'arf\'as' theorem by any
``expansive'' -uniform hypergraph on vertices (that is, a -uniform
hypergraph on vertices in which in which for all
disjoint sets with for all ), then one gets a largest monochromatic component of essentially the same
size (within a small error term depending on and ). As corollaries
we recover a number of known results about large monochromatic components in
random hypergraphs and random Steiner triple systems, often with drastically
improved bounds on the error terms.
Gy\'arf\'as' result is equivalent to the dual problem of determining the
smallest maximum degree of an arbitrary -partite -uniform hypergraph with
edges in which every set of edges has a common intersection. In this
language, our result says that if one replaces the condition that every set of
edges has a common intersection with the condition that for every
collection of disjoint sets with
for all there exists for all
such that , then the maximum degree of
is essentially the same (within a small error term depending on and
). We prove our results in this dual setting.Comment: 18 page
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