70 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
Graph Relations and Constrained Homomorphism Partial Orders
We consider constrained variants of graph homomorphisms such as embeddings,
monomorphisms, full homomorphisms, surjective homomorpshims, and locally
constrained homomorphisms. We also introduce a new variation on this theme
which derives from relations between graphs and is related to
multihomomorphisms. This gives a generalization of surjective homomorphisms and
naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs.
Both R-cores and R-cocores of graphs are unique up to isomorphism and can be
computed in polynomial time.
The theory of the graph homomorphism order is well developed, and from it we
consider analogous notions defined for orders induced by constrained
homomorphisms. We identify corresponding cores, prove or disprove universality,
characterize gaps and dualities. We give a new and significantly easier proof
of the universality of the homomorphism order by showing that even the class of
oriented cycles is universal. We provide a systematic approach to simplify the
proofs of several earlier results in this area. We explore in greater detail
locally injective homomorphisms on connected graphs, characterize gaps and show
universality. We also prove that for every the homomorphism order on
the class of line graphs of graphs with maximum degree is universal
Homomorphism complexes, reconfiguration, and homotopy for directed graphs
The neighborhood complex of a graph was introduced by Lov\'asz to provide
topological lower bounds on chromatic number. More general homomorphism
complexes of graphs were further studied by Babson and Kozlov. Such `Hom
complexes' are also related to mixings of graph colorings and other
reconfiguration problems, as well as a notion of discrete homotopy for graphs.
Here we initiate the detailed study of Hom complexes for directed graphs
(digraphs). For any pair of digraphs graphs and , we consider the
polyhedral complex that parametrizes the directed graph
homomorphisms . Hom complexes of digraphs have applications
in the study of chains in graded posets and cellular resolutions of monomial
ideals. We study examples of directed Hom complexes and relate their
topological properties to certain graph operations including products,
adjunctions, and foldings. We introduce a notion of a neighborhood complex for
a digraph and prove that its homotopy type is recovered as the Hom complex of
homomorphisms from a directed edge. We establish a number of results regarding
the topology of directed neighborhood complexes, including the dependence on
directed bipartite subgraphs, a digraph version of the Mycielski construction,
as well as vanishing theorems for higher homology. The Hom complexes of
digraphs provide a natural framework for reconfiguration of homomorphisms of
digraphs. Inspired by notions of directed graph colorings we study the
connectivity of for a tournament. Finally, we use
paths in the internal hom objects of digraphs to define various notions of
homotopy, and discuss connections to the topology of Hom complexes.Comment: 34 pages, 10 figures; V2: some changes in notation, clarified
statements and proofs, other corrections and minor revisions incorporating
comments from referee
Algebraic and Computer-based Methods in the Undirected Degree/diameter Problem - a Brief Survey
This paper discusses the most popular algebraic techniques and computational methods that have been used to construct large graphs with given degree and diameter
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