2,965 research outputs found
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
A Survey on Monochromatic Connections of Graphs
The concept of monochromatic connection of graphs was introduced by Caro and
Yuster in 2011. Recently, a lot of results have been published about it. In
this survey, we attempt to bring together all the results that dealt with it.
We begin with an introduction, and then classify the results into the following
categories: monochromatic connection coloring of edge-version, monochromatic
connection coloring of vertex-version, monochromatic index, monochromatic
connection coloring of total-version.Comment: 26 pages, 3 figure
Covering graphs by monochromatic trees and Helly-type results for hypergraphs
How many monochromatic paths, cycles or general trees does one need to cover
all vertices of a given -edge-coloured graph ? These problems were
introduced in the 1960s and were intensively studied by various researchers
over the last 50 years. In this paper, we establish a connection between this
problem and the following natural Helly-type question in hypergraphs. Roughly
speaking, this question asks for the maximum number of vertices needed to cover
all the edges of a hypergraph if it is known that any collection of a few
edges of has a small cover. We obtain quite accurate bounds for the
hypergraph problem and use them to give some unexpected answers to several
questions about covering graphs by monochromatic trees raised and studied by
Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao,
Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi
Constrained Ramsey Numbers
For two graphs S and T, the constrained Ramsey number f(S, T) is the minimum
n such that every edge coloring of the complete graph on n vertices, with any
number of colors, has a monochromatic subgraph isomorphic to S or a rainbow
(all edges differently colored) subgraph isomorphic to T. The Erdos-Rado
Canonical Ramsey Theorem implies that f(S, T) exists if and only if S is a star
or T is acyclic, and much work has been done to determine the rate of growth of
f(S, T) for various types of parameters. When S and T are both trees having s
and t edges respectively, Jamison, Jiang, and Ling showed that f(S, T) <=
O(st^2) and conjectured that it is always at most O(st). They also mentioned
that one of the most interesting open special cases is when T is a path. In
this work, we study this case and show that f(S, P_t) = O(st log t), which
differs only by a logarithmic factor from the conjecture. This substantially
improves the previous bounds for most values of s and t.Comment: 12 pages; minor revision
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
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