27 research outputs found

    Extension from precoloured sets of edges

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    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

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    Let TT be a distinguished subset of vertices in a graph GG. A TT-\emph{Steiner tree} is a subgraph of GG that is a tree and that spans TT. Kriesell conjectured that GG contains kk pairwise edge-disjoint TT-Steiner trees provided that every edge-cut of GG that separates TT has size 2k\ge 2k. When T=V(G)T=V(G) a TT-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 2k2k is replaced by 24k24k, and recently West and Wu have lowered this value to 6.5k6.5k. Our main result makes a further improvement to 5k+45k+4.Comment: 38 pages, 4 figure

    The Parameterized Complexity Binary CSP for Graphs with a Small Vertex Cover and Related Results

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    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[2d+12d+1] with as parameter the size of a vertex modulator to graphs of treedepth cc, or forests of depth dd, for constant c1c\geq 1, W[tt]-hard for all tNt\in \mathbb{N} 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

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    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 kk-choosable if given lists of at least kk 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 Δ\Delta is cΔ2c\Delta^2-choosable, for some constant cc. We prove this result with c=1c=1 (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 kk is nonrepetitively O(k2)O(k^{2})-colourable.Comment: v4: Minor changes made following helpful comments by the referee

    Mixing graph colourings

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    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

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    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 K4K_4, 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

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    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 PkP_k-free graphs. In particular, we prove that k-COLOURING on P8P_8-free graphs is NP-complete, 4-PRECOLOURING EXTENSION P7P_7-free graphs is NP-complete, and LIST 4-COLOURING on P6P_6-free graphs is NP-complete. In addition, we show the existence of an integer r such that k-COLOURING is NP-complete for PrP_r-free graphs with girth 4. In contrast, we determine for any fixed girth g4g\geq 4 a lower bound r(g)r(g) such that every Pr(g)P_{r(g)}-free graph with girth at least gg 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 (P2+P3)(P_2+P_3)-free graphs, a polynomial-time algorithm for solving LIST 3-Colouring on (P2+P4)(P_2+P_4)-free graphs, and a polynomial-time algorithm for solving LIST 3-COLOURING on sP3sP_3-free graphs. We prove that LIST k-COLOURING for (Ks,t,Pr)(K_{s,t},P_r)-free graphs is also polynomial-time solvable. We obtain several new dichotomies by combining the above results with some known results
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