47 research outputs found

    Uniform hypergraphs

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    Defective and Clustered Graph Colouring

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    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" dd if each monochromatic component has maximum degree at most dd. A colouring has "clustering" cc if each monochromatic component has at most cc 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 KtK_t as a minor, graphs excluding Ks,tK_{s,t} as a minor, and graphs excluding an arbitrary graph HH 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

    Research problems

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    Topics in graph colouring and extremal graph theory

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    In this thesis we consider three problems related to colourings of graphs and one problem in extremal graph theory. Let GG be a connected graph with nn vertices and maximum degree Δ(G)\Delta(G). Let Rk(G)R_k(G) denote the graph with vertex set all proper kk-colourings of GG and two kk-colourings are joined by an edge if they differ on the colour of exactly one vertex. Our first main result states that RΔ(G)+1(G)R_{\Delta(G)+1}(G) has a unique non-trivial component with diameter O(n2)O(n^2). This result can be viewed as a reconfigurations analogue of Brooks' Theorem and completes the study of reconfigurations of colourings of graphs with bounded maximum degree. A Kempe change is the operation of swapping some colours aa, bb of a component of the subgraph induced by vertices with colour aa or bb. Two colourings are Kempe equivalent if one can be obtained from the other by a sequence of Kempe changes. Our second main result states that all Δ(G)\Delta(G)-colourings of a graph GG are Kempe equivalent unless GG is the complete graph or the triangular prism. This settles a conjecture of Mohar (2007). Motivated by finding an algorithmic version of a structure theorem for bull-free graphs due to Chudnovsky (2012), we consider the computational complexity of deciding if the vertices of a graph can be partitioned into two parts such that one part is triangle-free and the other part is a collection of complete graphs. We show that this problem is NP-complete when restricted to five classes of graphs (including bull-free graphs) while polynomial-time solvable for the class of cographs. Finally we consider a graph-theoretic version formulated by Holroyd, Spencer and Talbot (2007) of the famous Erd\H{o}s-Ko-Rado Theorem in extremal combinatorics and obtain some results for the class of trees

    A brief history of edge-colorings – with personal reminiscences

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    In this article we survey some important milestones in the history of edge-colorings of graphs, from the earliest contributions of Peter Guthrie Tait and Dénes König to very recent wor
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