75 research outputs found
Countable graphs are majority 3-choosable
The Unfriendly Partition Conjecture posits that every countable graph admits
a 2-colouring in which for each vertex there are at least as many bichromatic
edges containing that vertex as monochromatic ones. This is not known in
general, but it is known that a 3-colouring with this property always exists.
Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this
conjecture, and proved that such a colouring exists for lists of size 4. We
improve their result to lists of size 3; the proof extends to directed acyclic
graphs. We also discuss some generalisations.Comment: 6 pages. Minor changes including adding a referenc
On generalised majority edge-colourings of graphs
A -majority -edge-colouring of a graph is a colouring of
its edges with colours such that for every colour and each vertex
of , at most 'th of the edges incident with have colour
. We conjecture that for every integer , each graph with minimum
degree is -majority -edge-colourable and
observe that such result would be best possible. This was already known to hold
for . We support the conjecture by proving it with instead of
, which confirms the right order of magnitude of the conjectured optimal
lower bound for . We at the same time improve the previously known
bound of order , based on a straightforward probabilistic approach.
As this technique seems not applicable towards any further improvement, we use
a more direct non-random approach. We also strengthen our result, in particular
substituting by . Finally, we provide the proof
of the conjecture itself for and completely solve an analogous
problem for the family of bipartite graphs.Comment: 18 page
Countable graphs are majority 3-choosable
The Unfriendly Partition Conjecture posits that every countable graph admits a -colouring in which for each vertex there are at least as many bichromatic edges containing that vertex as monochromatic ones. This is not known in general, but it is known that a -colouring with this property always exists. Anholcer, Bosek and Grytczuk recently gave a list-colouring version of this conjecture, and proved that such a colouring exists for lists of size . We improve their result to lists of size ; the proof extends to directed acyclic graphs. We also discuss some generalisations
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
Combinatorics
Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their
properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic
and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization,
Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions.
This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session
Recommended from our members
Extremal and Structural Problems of Graphs
In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values.
We begin in Chapter~ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance . The main result of Chapter~ comes close to proving this conjecture. Moreover, in Chapter~, we completely answer the previous question for the class of planar graphs.
Next, in Chapter~, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph must have to guarantee that, for any two-colouring of , we can partition into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours.
The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~. Given a graph and a set of colours (for some integer ), we define to be the minimum number of -coloured edges in a graph on vertices which does not contain a rainbow copy of but the addition of any non-edge in any colour from creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of , as a function of , for every connected graph of minimum degree greater than and for every integer .
In Chapter~, we consider the following question: under what conditions does a Hamiltonian graph on vertices possess a second cycle of length at least ?
We prove that the `weak' assumption of a minimum degree greater or equal to guarantees the existence of such a long cycle.
We solve two problems related to majority colouring in Chapter~. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number , the smallest positive integer such that every digraph can be coloured with colours, where each vertex has the same colour as at most a proportion of of its out-neighbours. Our main theorem states that .
We study the following problem, raised by Caro and Yuster, in Chapter~. Does every graph contain a `large' induced subgraph which has vertices of degree exactly ? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every , there exists such that any vertex graph with maximum degree contains an induced subgraph with at least vertices such that contains at least vertices of the same degree . This result is sharp up to the order of .
%Subsequently, we investigate a concept called . A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on vertices must possess a vertex of degree linear in . Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus.
Finally, in Chapter~, we move on to examine -linked tournaments. A tournament is said to be -linked if for any two disjoint sets of vertices and there are directed vertex disjoint paths such that joins to for . We prove that any strongly-connected tournament with sufficiently large minimum out-degree is -linked. This result comes close to proving a conjecture of Pokrovskiy
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