2,504 research outputs found
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
On the structure of complete graphs without alternating cycles
AbstractThe structure of edge-coloured complete graphs Kp which do not contain cycles with adjacent lines differently coloured is completely characterized. This characterization further enables us to prove that evey such Kp (p⩾2) has exactly one connected monochromatic spanning subgraph. Moreover, assume that edges in Kp are coloured by k colours so that no vertex is on more than Δ edges of each colour and less than δ edges of each colour. If such a Kp exists then p⩽Δ+1 and p⩾kδ+1, whereas, any one of these inequalities will ensure the existence of such a Kp
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
Note on alternating directed cycles
The problem of the existence of an alternating simple dicycle in a 2-arc-coloured digraph is considered. This is a generalization of the alternating cycle problem in 2-edge-coloured graphs and the even dicycle problem (both are polynomial time solvable). We prove that the alternating dicycle problem is NP -complete. Let f(n)(g(n), resp.) be the minimum integer such that if every monochromatic indegree and outdegree in a strongly connected 2-arc-coloured digraph (any 2-arc-coloured digraph, resp.) D is at least f(n)(g(n), resp.), then D has an alternating simple dicycle. We show that f(n) = #(log n) and g(n) = #(log n). ? 1998 Elsevier Science B.V. All rights reserved Keywords: Alternating cycles; Even cycles; Edge-coloured directed graph 1. Introduction, terminology and notation We shall assume that the reader is familiar with the standard terminology on graphs and digraphs and refer the reader to [4]. We consider digraphs without loops and multiple arcs. The arcs of digraphs are colo..
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
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 local edge colourings
An edge colouring of a graph is said to be an -local colouring if the
edges incident to any vertex are coloured with at most colours.
Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of
any -locally coloured complete graph may be partitioned into two disjoint
monochromatic cycles of different colours. Moreover, for any natural number
, we show that the vertex set of any -locally coloured complete graph may
be partitioned into disjoint monochromatic cycles. This
generalises a result of Erd\H{o}s, Gy\'arf\'as and Pyber.Comment: 10 page
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
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