24 research outputs found
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
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
Monochromatic cycle power partitions
Improving our earlier result we show that for every integer k≥1 there exists a c(k) such that in every 2-colored complete graph apart from at most c(k) vertices the vertex set can be covered by 200k2logk vertex disjoint monochromatic kth powers of cycles. © 2016 Elsevier B.V
Partitioning edge-coloured complete graphs into monochromatic cycles and paths
A conjecture of Erd\H{o}s, Gy\'arf\'as, and Pyber says that in any
edge-colouring of a complete graph with r colours, it is possible to cover all
the vertices with r vertex-disjoint monochromatic cycles. So far, this
conjecture has been proven only for r = 2. In this paper we show that in fact
this conjecture is false for all r > 2. In contrast to this, we show that in
any edge-colouring of a complete graph with three colours, it is possible to
cover all the vertices with three vertex-disjoint monochromatic paths, proving
a particular case of a conjecture due to Gy\'arf\'as. As an intermediate result
we show that in any edge-colouring of the complete graph with the colours red
and blue, it is possible to cover all the vertices with a red path, and a
disjoint blue balanced complete bipartite graph.Comment: 25 pages, 3 figure
Partitioning edge-colored hypergraphs into few monochromatic tight cycles
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmiklóssy [Discrete Math., 340 (2017), pp. 2053-2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number
Partitioning Edge-Colored Hypergraphs into Few Monochromatic Tight Cycles
Confirming a conjecture of Gy´arf´as, we prove that, for all natural numbers k and
r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a
bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We
further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete
graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes,
Soukup, Soukup, and Szentmikl´ossy [Discrete Math., 340 (2017), pp. 2053–2069]. In fact we prove a
common generalization of both theorems which further extends these results to all host hypergraphs
of bounded independence number