249 research outputs found

    Subgraphs and Colourability of Locatable Graphs

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    We study a game of pursuit and evasion introduced by Seager in 2012, in which a cop searches the robber from outside the graph, using distance queries. A graph on which the cop wins is called locatable. In her original paper, Seager asked whether there exists a characterisation of the graph property of locatable graphs by either forbidden or forbidden induced subgraphs, both of which we answer in the negative. We then proceed to show that such a characterisation does exist for graphs of diameter at most 2, stating it explicitly, and note that this is not true for higher diameter. Exploring a different direction of topic, we also start research in the direction of colourability of locatable graphs, we also show that every locatable graph is 4-colourable, but not necessarily 3-colourable.Comment: 25 page

    The degree-diameter problem for sparse graph classes

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    The degree-diameter problem asks for the maximum number of vertices in a graph with maximum degree Δ\Delta and diameter kk. For fixed kk, the answer is Θ(Δk)\Theta(\Delta^k). We consider the degree-diameter problem for particular classes of sparse graphs, and establish the following results. For graphs of bounded average degree the answer is Θ(Δk1)\Theta(\Delta^{k-1}), and for graphs of bounded arboricity the answer is \Theta(\Delta^{\floor{k/2}}), in both cases for fixed kk. For graphs of given treewidth, we determine the the maximum number of vertices up to a constant factor. More precise bounds are given for graphs of given treewidth, graphs embeddable on a given surface, and apex-minor-free graphs

    Nonrepetitive Colourings of Planar Graphs with O(logn)O(\log n) Colours

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    A vertex colouring of a graph is \emph{nonrepetitive} if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The \emph{nonrepetitive chromatic number} of a graph GG is the minimum integer kk such that GG has a nonrepetitive kk-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O(n)O(\sqrt{n}) for nn-vertex planar graphs. We prove a O(logn)O(\log n) upper bound

    A Study of kk-dipath Colourings of Oriented Graphs

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    We examine tt-colourings of oriented graphs in which, for a fixed integer k1k \geq 1, vertices joined by a directed path of length at most kk must be assigned different colours. A homomorphism model that extends the ideas of Sherk for the case k=2k=2 is described. Dichotomy theorems for the complexity of the problem of deciding, for fixed kk and tt, whether there exists such a tt-colouring are proved.Comment: 14 page

    The Complexity of Surjective Homomorphism Problems -- a Survey

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    We survey known results about the complexity of surjective homomorphism problems, studied in the context of related problems in the literature such as list homomorphism, retraction and compaction. In comparison with these problems, surjective homomorphism problems seem to be harder to classify and we examine especially three concrete problems that have arisen from the literature, two of which remain of open complexity

    Recognizing Graphs Close to Bipartite Graphs with an Application to Colouring Reconfiguration

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    We continue research into a well-studied family of problems that ask whether the vertices of a graph can be partitioned into sets AA and~BB, where AA is an independent set and BB induces a graph from some specified graph class G{\cal G}. We let G{\cal G} be the class of kk-degenerate graphs. This problem is known to be polynomial-time solvable if k=0k=0 (bipartite graphs) and NP-complete if k=1k=1 (near-bipartite graphs) even for graphs of maximum degree 44. Yang and Yuan [DM, 2006] showed that the k=1k=1 case is polynomial-time solvable for graphs of maximum degree 33. This also follows from a result of Catlin and Lai [DM, 1995]. We consider graphs of maximum degree k+2k+2 on nn vertices. We show how to find AA and BB in O(n)O(n) time for k=1k=1, and in O(n2)O(n^2) time for k2k\geq 2. Together, these results provide an algorithmic version of a result of Catlin [JCTB, 1979] and also provide an algorithmic version of a generalization of Brook's Theorem, which was proven in a more general way by Borodin, Kostochka and Toft [DM, 2000] and Matamala [JGT, 2007]. Moreover, the two results enable us to complete the complexity classification of an open problem of Feghali et al. [JGT, 2016]: finding a path in the vertex colouring reconfiguration graph between two given \ell-colourings of a graph of maximum degree kk

    Colourings of cubic graphs inducing isomorphic monochromatic subgraphs

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    A kk-bisection of a bridgeless cubic graph GG is a 22-colouring of its vertex set such that the colour classes have the same cardinality and all connected components in the two subgraphs induced by the colour classes (monochromatic components in what follows) have order at most kk. Ban and Linial conjectured that every bridgeless cubic graph admits a 22-bisection except for the Petersen graph. A similar problem for the edge set of cubic graphs has been studied: Wormald conjectured that every cubic graph GG with E(G)0(mod2)|E(G)| \equiv 0 \pmod 2 has a 22-edge colouring such that the two monochromatic subgraphs are isomorphic linear forests (i.e. a forest whose components are paths). Finally, Ando conjectured that every cubic graph admits a bisection such that the two induced monochromatic subgraphs are isomorphic. In this paper, we give a detailed insight into the conjectures of Ban-Linial and Wormald and provide evidence of a strong relation of both of them with Ando's conjecture. Furthermore, we also give computational and theoretical evidence in their support. As a result, we pose some open problems stronger than the above mentioned conjectures. Moreover, we prove Ban-Linial's conjecture for cubic cycle permutation graphs. As a by-product of studying 22-edge colourings of cubic graphs having linear forests as monochromatic components, we also give a negative answer to a problem posed by Jackson and Wormald about certain decompositions of cubic graphs into linear forests.Comment: 33 pages; submitted for publicatio
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