27 research outputs found
Extension from precoloured sets of edges
We consider precolouring extension problems for proper edge-colourings of graphs and multigraphs, in an attempt to prove stronger versions of Vizing's and Shannon's bounds on the chromatic index of (multi)graphs in terms of their maximum degree Δ. We are especially interested in the following question: when is it possible to extend a precoloured matching to a colouring of all edges of a (multi)graph? This question turns out to be related to the notorious List Colouring Conjecture and other classic notions of choosability
Packing Steiner Trees
Let be a distinguished subset of vertices in a graph . A
-\emph{Steiner tree} is a subgraph of that is a tree and that spans .
Kriesell conjectured that contains pairwise edge-disjoint -Steiner
trees provided that every edge-cut of that separates has size .
When a -Steiner tree is a spanning tree and the conjecture is a
consequence of a classic theorem due to Nash-Williams and Tutte. Lau proved
that Kriesell's conjecture holds when is replaced by , and recently
West and Wu have lowered this value to . Our main result makes a further
improvement to .Comment: 38 pages, 4 figure
The Parameterized Complexity Binary CSP for Graphs with a Small Vertex Cover and Related Results
In this paper, we show that Binary CSP with the size of a vertex cover as
parameter is complete for the class W[3]. We obtain a number of related results
with variations of the proof techniques, that include: Binary CSP is complete
for W[] with as parameter the size of a vertex modulator to graphs of
treedepth , or forests of depth , for constant , W[]-hard for
all with treewidth as parameter, and hard for W[SAT] with
feedback vertex set as parameter. As corollaries, we give some hardness and
membership problems for classes in the W-hierarchy for List Colouring under
different parameterisations
Nonrepetitive Colouring via Entropy Compression
A vertex colouring of a graph is \emph{nonrepetitive} if there is no path
whose first half receives the same sequence of colours as the second half. A
graph is nonrepetitively -choosable if given lists of at least colours
at each vertex, there is a nonrepetitive colouring such that each vertex is
coloured from its own list. It is known that every graph with maximum degree
is -choosable, for some constant . We prove this result
with (ignoring lower order terms). We then prove that every subdivision
of a graph with sufficiently many division vertices per edge is nonrepetitively
5-choosable. The proofs of both these results are based on the Moser-Tardos
entropy-compression method, and a recent extension by Grytczuk, Kozik and Micek
for the nonrepetitive choosability of paths. Finally, we prove that every graph
with pathwidth is nonrepetitively -colourable.Comment: v4: Minor changes made following helpful comments by the referee
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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
Mixing graph colourings
This thesis investigates some problems related to graph colouring, or, more precisely, graph re-colouring. Informally, the basic question addressed can be phrased as follows. Suppose one is given a graph G whose vertices can be properly k-coloured, for some k ≥ 2. Is it possible to transform any k-colouring of G into any other by recolouring vertices of G one at a time, making sure a proper k-colouring of G is always maintained? If the answer is in the affirmative, G is said to be k-mixing. The related problem of deciding whether, given two k-colourings of G, it is possible to transform one into the other by recolouring vertices one at a time, always maintaining a proper k-colouring of G, is also considered.
These questions can be considered as having a bearing on certain mathematical and ‘real-world’ problems. In particular, being able to recolour any colouring of a given graph to any other colouring is a necessary pre-requisite for the method of sampling colourings known as Glauber dynamics. The results presented in this thesis may also find application in the context of frequency reassignment: given that the problem of assigning radio frequencies in a wireless communications network is often modelled as a graph colouring problem, the task of re-assigning frequencies in such a network can be thought of as a graph recolouring problem.
Throughout the thesis, the emphasis is on the algorithmic aspects and the computational complexity of the questions described above. In other words, how easily, in terms of computational resources used, can they be answered? Strong results are obtained for the k = 3 case of the first question, where a characterisation theorem for 3-mixing graphs is given. For the second question, a dichotomy theorem for the complexity of the problem is proved: the problem is solvable in polynomial time for k ≤ 3 and PSPACE-complete for k ≥ 4. In addition, the possible length of a shortest sequence of recolourings between two colourings is investigated, and an interesting connection between the tractability of the problem and its underlying structure is established. Some variants of the above problems are also explored
Complexity of colouring problems restricted to unichord-free and \{square,unichord\}-free graphs
A \emph{unichord} in a graph is an edge that is the unique chord of a cycle.
A \emph{square} is an induced cycle on four vertices. A graph is
\emph{unichord-free} if none of its edges is a unichord. We give a slight
restatement of a known structure theorem for unichord-free graphs and use it to
show that, with the only exception of the complete graph , every
square-free, unichord-free graph of maximum degree~3 can be total-coloured with
four colours. Our proof can be turned into a polynomial time algorithm that
actually outputs the colouring. This settles the class of square-free,
unichord-free graphs as a class for which edge-colouring is NP-complete but
total-colouring is polynomial
Graph Colouring with Input Restrictions
In this thesis, we research the computational complexity of the graph colouring problem and its variants including precolouring extension and list colouring for graph classes that can be characterised by forbidding one or more induced subgraphs. We investigate the structural properties of such graph classes and prove a number of new properties. We then consider to what extent these properties can be used for efficiently solving the three types of colouring problems on these graph classes. In some cases we obtain polynomial-time algorithms, whereas other cases turn out to be NP-complete.
We determine the computational complexity of k-COLOURING, k-PRECOLOURING EXTENSION and LIST k-COLOURING on -free graphs. In particular, we prove that k-COLOURING on -free graphs is NP-complete, 4-PRECOLOURING EXTENSION -free graphs is NP-complete, and LIST 4-COLOURING on -free graphs is NP-complete. In addition, we show the existence of an integer r such that k-COLOURING is NP-complete for -free graphs with girth 4. In contrast, we determine for any fixed girth a lower bound such that every -free graph with girth at least is 3-colourable. We also prove that 3-LIST COLOURING is NP-complete for complete graphs minus a matching. We present a polynomial-time algorithm for solving 4-PRECOLOURING EXTENSION on -free graphs, a polynomial-time algorithm for solving LIST 3-Colouring on -free graphs, and a polynomial-time algorithm for solving LIST 3-COLOURING on -free graphs. We prove that LIST k-COLOURING for -free graphs is also polynomial-time solvable. We obtain several new dichotomies by combining the above results with some known results