965 research outputs found
Local Perspectives on Planar Colouring
In 1994, Thomassen famously proved that every planar graph is 5-choosable, resolving a conjecture initially posed by Vizing and, independently, Erdos, Rubin, and Taylor in the 1970s. Later, Thomassen proved that every planar graph of girth at least five is 3-choosable. In this thesis, we introduce the concept of a local girth list assignment: a list assignment wherein the list size of a vertex depends not on the girth of the graph, but rather on the length of the shortest cycle in which the vertex is contained. We state and prove a local list colouring theorem unifying the two theorems of Thomassen mentioned above. In particular, we show that if G is a planar graph and L is a list assignment for G such that |L(v)| ≥ 3 for all v in V(G); |L(v)| ≥ 4 for every vertex v contained in a 4-cycle; and |L(v)| ≥ 5 for every vertex v contained in a triangle, then G admits an L-colouring.
Next, we generalize a framework of list colouring results to correspondence colouring. Correspondence colouring is a generalization of list colouring wherein we localize the meaning of the colours available to each vertex. As pointed out by Dvorak and Postle, both of Thomassen's theorems on the 5-choosability of planar graphs and 3-choosability of planar graphs of girth at least five carry over to the correspondence colouring setting. In this thesis, we show that the family of graphs that are critical for 5-correspondence colouring as well as the family of graphs of girth at least five that are critical for 3-correspondence colouring form hyperbolic families. Analogous results for list colouring were shown by Postle and Thomas. Using results on hyperbolic families proved by Postle and Thomas, we show further that this implies that locally planar graphs are 5-correspondence colourable; and, using results of Dvorak and Kawarabayashi, that there exist linear-time algorithms for the decidability of 5-correspondence colouring for embedded graphs. We show analogous results for 3-correspondence colouring graphs of girth at least five.
Finally we show that, in general, slightly stronger hyperbolicity theorems imply that the associated family of planar graphs have exponentially many colourings. The existence of exponentially many colourings has been studied before for list-colouring: for instance, Thomassen showed (without using hyperbolicity) that planar graphs have exponentially many 5-list colourings, and that planar graphs of girth at least five have exponentially many 3-list colourings. Using our stronger hyperbolicity theorems, we prove that planar graphs of girth at least five have exponentially many 3-correspondence colourings, and that planar graphs have exponentially many 5-correspondence colourings. This latter result proves a conjecture of Langhede and Thomassen. As correspondence colouring generalizes list colouring, our theorems also provide new, independent proofs that there are exponentially many 5-list colourings of planar graphs, and 3-list colourings of planar graphs of girth at least five
Defective and Clustered Graph Colouring
Consider the following two ways to colour the vertices of a graph where the
requirement that adjacent vertices get distinct colours is relaxed. A colouring
has "defect" if each monochromatic component has maximum degree at most
. A colouring has "clustering" if each monochromatic component has at
most vertices. This paper surveys research on these types of colourings,
where the first priority is to minimise the number of colours, with small
defect or small clustering as a secondary goal. List colouring variants are
also considered. The following graph classes are studied: outerplanar graphs,
planar graphs, graphs embeddable in surfaces, graphs with given maximum degree,
graphs with given maximum average degree, graphs excluding a given subgraph,
graphs with linear crossing number, linklessly or knotlessly embeddable graphs,
graphs with given Colin de Verdi\`ere parameter, graphs with given
circumference, graphs excluding a fixed graph as an immersion, graphs with
given thickness, graphs with given stack- or queue-number, graphs excluding
as a minor, graphs excluding as a minor, and graphs excluding
an arbitrary graph as a minor. Several open problems are discussed.Comment: This is a preliminary version of a dynamic survey to be published in
the Electronic Journal of Combinatoric
The Complexity of Change
Many combinatorial problems can be formulated as "Can I transform
configuration 1 into configuration 2, if certain transformations only are
allowed?". An example of such a question is: given two k-colourings of a graph,
can I transform the first k-colouring into the second one, by recolouring one
vertex at a time, and always maintaining a proper k-colouring? Another example
is: given two solutions of a SAT-instance, can I transform the first solution
into the second one, by changing the truth value one variable at a time, and
always maintaining a solution of the SAT-instance? Other examples can be found
in many classical puzzles, such as the 15-Puzzle and Rubik's Cube.
In this survey we shall give an overview of some older and more recent work
on this type of problem. The emphasis will be on the computational complexity
of the problems: how hard is it to decide if a certain transformation is
possible or not?Comment: 28 pages, 6 figure
Disproof of the List Hadwiger Conjecture
The List Hadwiger Conjecture asserts that every -minor-free graph is
-choosable. We disprove this conjecture by constructing a
-minor-free graph that is not -choosable for every integer
Defective and Clustered Choosability of Sparse Graphs
An (improper) graph colouring has "defect" if each monochromatic subgraph
has maximum degree at most , and has "clustering" if each monochromatic
component has at most vertices. This paper studies defective and clustered
list-colourings for graphs with given maximum average degree. We prove that
every graph with maximum average degree less than is
-choosable with defect . This improves upon a similar result by Havet and
Sereni [J. Graph Theory, 2006]. For clustered choosability of graphs with
maximum average degree , no bound on the number of colours
was previously known. The above result with solves this problem. It
implies that every graph with maximum average degree is
-choosable with clustering 2. This extends a
result of Kopreski and Yu [Discrete Math., 2017] to the setting of
choosability. We then prove two results about clustered choosability that
explore the trade-off between the number of colours and the clustering. In
particular, we prove that every graph with maximum average degree is
-choosable with clustering , and is
-choosable with clustering . As an
example, the later result implies that every biplanar graph is 8-choosable with
bounded clustering. This is the best known result for the clustered version of
the earth-moon problem. The results extend to the setting where we only
consider the maximum average degree of subgraphs with at least some number of
vertices. Several applications are presented
On the Complexity of Role Colouring Planar Graphs, Trees and Cographs
We prove several results about the complexity of the role colouring problem.
A role colouring of a graph is an assignment of colours to the vertices of
such that two vertices of the same colour have identical sets of colours in
their neighbourhoods. We show that the problem of finding a role colouring with
colours is NP-hard for planar graphs. We show that restricting the
problem to trees yields a polynomially solvable case, as long as is either
constant or has a constant difference with , the number of vertices in the
tree. Finally, we prove that cographs are always -role-colourable for
and construct such a colouring in polynomial time
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