59 research outputs found
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of eïŹcient solutions and algorithms for computationally diïŹcult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
Uniquely D-colourable digraphs with large girth
Let C and D be digraphs. A mapping is a C-colouring if for
every arc of D, either is an arc of C or , and the
preimage of every vertex of C induces an acyclic subdigraph in D. We say that D
is C-colourable if it admits a C-colouring and that D is uniquely C-colourable
if it is surjectively C-colourable and any two C-colourings of D differ by an
automorphism of C. We prove that if a digraph D is not C-colourable, then there
exist digraphs of arbitrarily large girth that are D-colourable but not
C-colourable. Moreover, for every digraph D that is uniquely D-colourable,
there exists a uniquely D-colourable digraph of arbitrarily large girth. In
particular, this implies that for every rational number , there are
uniquely circularly r-colourable digraphs with arbitrarily large girth.Comment: 21 pages, 0 figures To be published in Canadian Journal of
Mathematic
Digraphs and homomorphisms: Cores, colorings, and constructions
A natural digraph analogue of the graph-theoretic concept of an `independent set\u27 is that of an acyclic set, namely a set of vertices not spanning a directed cycle. Hence a digraph analogue of a graph coloring is a decomposition of the vertex set into acyclic sets
Colorings of graphs, digraphs, and hypergraphs
Brooks' Theorem ist eines der bekanntesten Resultate ĂŒber GraphenfĂ€rbungen: Sei G ein zusammenhĂ€ngender Graph mit Maximalgrad d. Ist G kein vollstĂ€ndiger Graph, so lassen sich die Ecken von G so mit d Farben fĂ€rben, dass zwei benachbarte Ecken unterschiedlich gefĂ€rbt sind. In der vorliegenden Arbeit liegt der Fokus auf Verallgemeinerungen von Brooks Theorem fĂŒr FĂ€rbungen von Hypergraphen und gerichteten Graphen. Eine FĂ€rbung eines Hypergraphen ist eine FĂ€rbung der Ecken so, dass keine Kante monochromatisch ist. Auf Hypergraphen erweitert wurde der Satz von Brooks von R.P. Jones. Im ersten Teil der Dissertation werden Möglichkeiten aufgezeigt, das Resultat von Jones weiter zu verallgemeinern. KernstĂŒck ist ein Zerlegungsresultat: Zu einem Hypergraphen H und einer Folge f=(f_1,âŠ,f_p) von Funktionen, welche von V(H) in die natĂŒrlichen Zahlen abbilden, wird untersucht, ob es eine Zerlegung von H in induzierte Unterhypergraphen H_1,âŠ,H_p derart gibt, dass jedes H_i strikt f_i-degeneriert ist. Dies bedeutet, dass jeder Unterhypergraph H_i' von H_i eine Ecke v enthĂ€lt, deren Grad in H_i' kleiner als f_i(v) ist. Es wird bewiesen, dass die Bedingung f_1(v)+âŠ+f_p(v) \geq d_H(v) fĂŒr alle v fast immer ausreichend fĂŒr die Existenz einer solchen Zerlegung ist und gezeigt, dass sich die AusnahmefĂ€lle gut charakterisieren lassen. Durch geeignete Wahl der Funktion f lassen sich viele bekannte Resultate ableiten, was im dritten Kapitel erörtert wird. Danach werden zwei weitere Verallgemeinerungen des Satzes von Jones bewiesen: Ein Theorem zu DP-FĂ€rbungen von Hypergraphen und ein Resultat, welches die chromatische Zahl eines Hypergraphen mit dessen maximalem lokalen Kantenzusammenhang verbindet. Der zweite Teil untersucht FĂ€rbungen gerichteter Graphen. Eine azyklische FĂ€rbung eines gerichteten Graphen ist eine FĂ€rbung der Eckenmenge des gerichteten Graphen sodass es keine monochromatischen gerichteten Kreise gibt. Auf dieses Konzept lassen sich viele klassische FĂ€rbungsresultate ĂŒbertragen. Dazu zĂ€hlt auch Brooks Theorem, wie von Mohar bewiesen wurde. Im siebten Kapitel werden DP-FĂ€rbungen gerichteter Graphen untersucht. Insbesondere erfolgt der Transfer von Mohars Theorem auf DP-FĂ€rbungen. Das darauffolgende Kapitel befasst sich mit kritischen gerichteten Graphen. Insbesondere werden Konstruktionen fĂŒr diese angegeben und die gerichtete Version des Satzes von HajĂłs bewiesen.Brooksâ Theorem is one of the most known results in graph coloring theory: Let G be a connected graph with maximum degree d >2. If G is not a complete graph, then there is a coloring of the vertices of G with d colors such that no two adjacent vertices get the same color. Based on Brooks' result, various research topics in graph coloring arose. Also, it became evident that Brooks' Theorem could be transferred to many other coloring-concepts. The present thesis puts its focus especially on two of those concepts: hypergraphs and digraphs. A coloring of a hypergraph H is a coloring of its vertices such that no edge is monochromatic. Brooks' Theorem for hypergraphs was obtained by R.P. Jones. In the first part of this thesis, we present several ways how to further extend Jones' theorem. The key element is a partition result, to which the second chapter is devoted. Given a hypergraph H and a sequence f=(f_1,âŠ,f_p) of functions, we examine if there is a partition of into induced subhypergraphs H_1,âŠ,H_p such that each of the H_i is strictly f_i-degenerate. This means that in each non-empty subhypergraph H_i' of H_i there is a vertex v having degree d_{H_i'}(v
Reconfiguring Graph Homomorphisms on the Sphere
Given a loop-free graph , the reconfiguration problem for homomorphisms to
(also called -colourings) asks: given two -colourings of of a
graph , is it possible to transform into by a sequence of
single-vertex colour changes such that every intermediate mapping is an
-colouring? This problem is known to be polynomial-time solvable for a wide
variety of graphs (e.g. all -free graphs) but only a handful of hard
cases are known. We prove that this problem is PSPACE-complete whenever is
a -free quadrangulation of the -sphere (equivalently, the plane)
which is not a -cycle. From this result, we deduce an analogous statement
for non-bipartite -free quadrangulations of the projective plane. This
include several interesting classes of graphs, such as odd wheels, for which
the complexity was known, and -chromatic generalized Mycielski graphs, for
which it was not.
If we instead consider graphs and with loops on every vertex (i.e.
reflexive graphs), then the reconfiguration problem is defined in a similar way
except that a vertex can only change its colour to a neighbour of its current
colour. In this setting, we use similar ideas to show that the reconfiguration
problem for -colourings is PSPACE-complete whenever is a reflexive
-free triangulation of the -sphere which is not a reflexive triangle.
This proof applies more generally to reflexive graphs which, roughly speaking,
resemble a triangulation locally around a particular vertex. This provides the
first graphs for which -Recolouring is known to be PSPACE-complete for
reflexive instances.Comment: 22 pages, 9 figure
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