6 research outputs found

    Boxicity and topological invariants

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    The boxicity of a graph G=(V,E)G=(V,E) is the smallest integer kk for which there exist kk interval graphs Gi=(V,Ei)G_i=(V,E_i), 1ik1 \le i \le k, such that E=E1EkE=E_1 \cap \cdots \cap E_k. In the first part of this note, we prove that every graph on mm edges has boxicity O(mlogm)O(\sqrt{m \log m}), which is asymptotically best possible. We use this result to study the connection between the boxicity of graphs and their Colin de Verdi\`ere invariant, which share many similarities. Known results concerning the two parameters suggest that for any graph GG, the boxicity of GG is at most the Colin de Verdi\`ere invariant of GG, denoted by μ(G)\mu(G). We observe that every graph GG has boxicity O(μ(G)4(logμ(G))2)O(\mu(G)^4(\log \mu(G))^2), while there are graphs GG with boxicity Ω(μ(G)logμ(G))\Omega(\mu(G)\sqrt{\log \mu(G)}). In the second part of this note, we focus on graphs embeddable on a surface of Euler genus gg. We prove that these graphs have boxicity O(glogg)O(\sqrt{g}\log g), while some of these graphs have boxicity Ω(glogg)\Omega(\sqrt{g \log g}). This improves the previously best known upper and lower bounds. These results directly imply a nearly optimal bound on the dimension of the adjacency poset of graphs on surfaces.Comment: 6 page

    Box representations of embedded graphs

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    A dd-box is the cartesian product of dd intervals of R\mathbb{R} and a dd-box representation of a graph GG is a representation of GG as the intersection graph of a set of dd-boxes in Rd\mathbb{R}^d. It was proved by Thomassen in 1986 that every planar graph has a 3-box representation. In this paper we prove that every graph embedded in a fixed orientable surface, without short non-contractible cycles, has a 5-box representation. This directly implies that there is a function ff, such that in every graph of genus gg, a set of at most f(g)f(g) vertices can be removed so that the resulting graph has a 5-box representation. We show that such a function ff can be made linear in gg. Finally, we prove that for any proper minor-closed class F\mathcal{F}, there is a constant c(F)c(\mathcal{F}) such that every graph of F\mathcal{F} without cycles of length less than c(F)c(\mathcal{F}) has a 3-box representation, which is best possible.Comment: 16 pages, 6 figures - revised versio

    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

    Coloring planar graphs with three colors and no large monochromatic components

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    We prove the existence of a function f:NNf :\mathbb{N} \to \mathbb{N} such that the vertices of every planar graph with maximum degree Δ\Delta can be 3-colored in such a way that each monochromatic component has at most f(Δ)f(\Delta) vertices. This is best possible (the number of colors cannot be reduced and the dependence on the maximum degree cannot be avoided) and answers a question raised by Kleinberg, Motwani, Raghavan, and Venkatasubramanian in 1997. Our result extends to graphs of bounded genus.Comment: v3: fixed a notation issue in Section

    Improper colourings inspired by Hadwiger’s conjecture

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    Hadwiger’s Conjecture asserts that every Kt-minor-free graph has a proper (t − 1)-colouring. We relax the conclusion in Hadwiger’s Conjecture via improper colourings. We prove that every Kt-minor-free graph is (2t − 2)-colourable with monochromatic components of order at most 1/2 (t − 2). This result has no more colours and much smaller monochromatic components than all previous results in this direction. We then prove that every Kt-minor-free graph is (t − 1)-colourable with monochromatic degree at most t − 2. This is the best known degree bound for such a result. Both these theorems are based on a decomposition method of independent interest. We give analogous results for Ks,t-minorfree graphs, which lead to improved bounds on generalised colouring numbers for these classes. Finally, we prove that graphs containing no Kt-immersion are 2-colourable with bounded monochromatic degree

    From the plane to higher surfaces

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    AbstractWe show that Grötzschʼs theorem extends to all higher surfaces in the sense that every triangle-free graph on a surface of Euler genus g becomes 3-colorable after deleting a set of at most 1000⋅g⋅f(g) vertices where f(g) is the smallest edge-width which guarantees a graph of Euler genus g and girth 5 to be 3-colorable.We derive this result from a general cutting technique which we also use to extend other results on planar graphs to higher surfaces in the same spirit, even after deleting only 1000g vertices. These include the 5-list-color theorem, results on arboricity, and various types of colorings, and decomposition theorems of planar graphs into two graphs with prescribed degeneracy properties.It is not known if the 4-color theorem extends in this way
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