15 research outputs found
Partitioning 2-edge-colored graphs by monochromatic paths and cycles
We present results on partitioning the vertices of -edge-colored graphs
into monochromatic paths and cycles. We prove asymptotically the two-color case
of a conjecture of S\'ark\"ozy: the vertex set of every -edge-colored graph
can be partitioned into at most monochromatic cycles, where
denotes the independence number of . Another direction, emerged
recently from a conjecture of Schelp, is to consider colorings of graphs with
given minimum degree. We prove that apart from vertices, the vertex
set of any -edge-colored graph with minimum degree at least
(1+\eps){3|V(G)|\over 4} can be covered by the vertices of two vertex
disjoint monochromatic cycles of distinct colors. Finally, under the assumption
that does not contain a fixed bipartite graph , we show that
in every -edge-coloring of , vertices can be covered by two
vertex disjoint paths of different colors, where is a constant depending
only on . In particular, we prove that , which is best possible
Vertex covering with monochromatic pieces of few colours
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the
edges of , there is a vertex cover by monochromatic paths of
the same colour, which is optimal up to a constant factor. The main goal of
this paper is to study the natural multi-colour generalization of this problem:
given two positive integers , what is the smallest number
such that in every colouring of the edges of with
colours, there exists a vertex cover of by
monochromatic paths using altogether at most different colours? For fixed
integers and as , we prove that , where is the chromatic number of
the Kneser gr aph . More generally, if one replaces by
an arbitrary -vertex graph with fixed independence number , then we
have , where this time around is the
chromatic number of the Kneser hypergraph . This
result is tight in the sense that there exist graphs with independence number
for which . This is in sharp
contrast to the case , where it follows from a result of S\'ark\"ozy
(2012) that depends only on and , but not on
the number of vertices. We obtain similar results for the situation where
instead of using paths, one wants to cover a graph with bounded independence
number by monochromatic cycles, or a complete graph by monochromatic
-regular graphs
Local colourings and monochromatic partitions in complete bipartite graphs
We show that for any -local colouring of the edges of the balanced
complete bipartite graph , its vertices can be covered with at
most~ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite -locally coloured graph with
disjoint monochromatic cycles.\\ We also determine the -local
bipartite Ramsey number of a path almost exactly: Every -local colouring of
the edges of contains a monochromatic path on vertices.Comment: 18 page
Large monochromatic components in edge colored graphs with a minimum degree condition
It is well-known that in every k-coloring of the edges of the complete graph Kn there is a monochromatic connected component of order at least (formula presented)k-1. In this paper we study an extension of this problem by replacing complete graphs by graphs of large minimum degree. For k = 2 the authors proved that δ(G) ≥(formula presented) ensures a monochromatic connected component with at least δ(G) + 1 vertices in every 2-coloring of the edges of a graph G with n vertices. This result is sharp, thus for k = 2 we really need a complete graph to guarantee that one of the colors has a monochromatic connected spanning subgraph. Our main result here is that for larger values of k the situation is different, graphs of minimum degree (1 − ϵk)n can replace complete graphs and still there is a monochromatic connected component of order at least (formula presented), in fact (formula presented) suffices. Our second result is an improvement of this bound for k = 3. If the edges of G with δ(G) ≥ (formula presented) are 3-colored, then there is a monochromatic component of order at least n/2. We conjecture that this can be improved to 9 and for general k we (onjectu) the following: if k ≥ 3 and G is a graph of order n such that δ(G) ≥ (formula presented) n, then in any k-coloring of the edges of G there is a monochromatic connected component of order at least (formula presented). © 2017, Australian National University. All rights reserved
Almost partitioning every -edge-coloured complete -graph into monochromatic tight cycles
A -uniform tight cycle is a -graph with a cyclic order of its vertices
such that every consecutive vertices from an edge. We show that for , every red-blue edge-coloured complete -graph on vertices contains
vertex-disjoint monochromatic tight cycles that together cover
vertices
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
Monochromatic cycle partitions in random graphs
Erd\H{o}s, Gy\'arf\'as and Pyber showed that every -edge-coloured complete
graph can be covered by vertex-disjoint monochromatic
cycles (independent of ). Here, we extend their result to the setting of
binomial random graphs. That is, we show that if , then with high probability any -edge-coloured
can be covered by at most vertex-disjoint monochromatic
cycles. This answers a question of Kor\'andi, Mousset, Nenadov, \v{S}kori\'{c}
and Sudakov.Comment: 16 pages, accepted in Combinatorics, Probability and Computin
Monochromatic cycle covers in random graphs
A classic result of Erd\H{o}s, Gy\'arf\'as and Pyber states that for every
coloring of the edges of with colors, there is a cover of its vertex
set by at most vertex-disjoint monochromatic cycles. In
particular, the minimum number of such covering cycles does not depend on the
size of but only on the number of colors. We initiate the study of this
phenomena in the case where is replaced by the random graph . Given a fixed integer and , we
show that with high probability the random graph has
the property that for every -coloring of the edges of , there is a
collection of monochromatic cycles covering all the
vertices of . Our bound on is close to optimal in the following sense:
if , then with high probability there are colorings of
such that the number of monochromatic cycles needed to
cover all vertices of grows with .Comment: 24 pages, 1 figure (minor changes, added figure