31,898 research outputs found

    Long cycles in 4-connected planar graphs

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    AbstractLet G be a 4-connected planar graph on n vertices. Malkevitch conjectured that if G contains a cycle of length 4, then G contains a cycle of length k for every kāˆˆ{n,nāˆ’1,ā€¦,3}. This conjecture is true for every kāˆˆ{n,nāˆ’1,ā€¦,nāˆ’6} with kā‰„3. In this paper, we prove that G also has a cycle of length nāˆ’7 provided nā‰„10

    Longer cycles in essentially 4-connected planar graphs

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    On longest cycles in essentially 4-connected planar graphs

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    A planar 3-connected graph G is essentially 4-connected if, for any 3-separator S of G, one component of the graph obtained from G by removing S is a single vertex. Jackson and Wormald proved that an essentially 4-connected planar graph on n vertices contains a cycle C such that . For a cubic essentially 4-connected planar graph G, GrĆ¼nbaum with Malkevitch, and Zhang showed that G has a cycle on at least Ā¾ n vertices. In the present paper the result of Jackson and Wormald is improved. Moreover, new lower bounds on the length of a longest cycle of G are presented if G is an essentially 4-connected planar graph of maximum degree 4 or G is an essentially 4-connected maximal planar graph

    Contact Representations of Graphs in 3D

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    We study contact representations of graphs in which vertices are represented by axis-aligned polyhedra in 3D and edges are realized by non-zero area common boundaries between corresponding polyhedra. We show that for every 3-connected planar graph, there exists a simultaneous representation of the graph and its dual with 3D boxes. We give a linear-time algorithm for constructing such a representation. This result extends the existing primal-dual contact representations of planar graphs in 2D using circles and triangles. While contact graphs in 2D directly correspond to planar graphs, we next study representations of non-planar graphs in 3D. In particular we consider representations of optimal 1-planar graphs. A graph is 1-planar if there exists a drawing in the plane where each edge is crossed at most once, and an optimal n-vertex 1-planar graph has the maximum (4n - 8) number of edges. We describe a linear-time algorithm for representing optimal 1-planar graphs without separating 4-cycles with 3D boxes. However, not every optimal 1-planar graph admits a representation with boxes. Hence, we consider contact representations with the next simplest axis-aligned 3D object, L-shaped polyhedra. We provide a quadratic-time algorithm for representing optimal 1-planar graph with L-shaped polyhedra

    Small cycle cover, group coloring with related problems

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    Bondy conjectured that if G is a simple 2-connected graph with n ā‰„ 3 vertices, then the edges of G can be covered by at most 2n-33 cycles. In Chapter 2, a result on small cycle cover is obtained and we also show that the result is as best as possible.;Thomassen conjectured that every 4-connected line graph is hamiltonian. In Chapters 3 and 4, we apply Catlin\u27s reduction method to study cycles in line graphs. Results about hamiltonian connectivity of line graphs and 3-edge-connected graphs are obtained. Several former results are extended.;Jaeger, Linial, Payan and Tarsi introduced group coloring in 1992 and proved that the group chromatic number for every planar graph is at most 6. It is shown that the bound 6 can be decreased to 5. Jaeger, Linial, Payan and Tarsi also proved that the group chromatic number for every planar graph with girth at least 4 is at most 4. Chapters 5 and 6 are devoted to the study of group coloring of graphs.;The concept of list coloring, choosability and choice number was introduced by Erdos, Rubin and Taylor in 1979 and Vizing in 1976. Alon and Tarsi proved that every bipartite planar graph is 3-choosable. Thomassen showed that every planar graph is 5-choosable and that every planar graph with girth at least 5 is 3-choosable. In Chapter 7, results on list coloring are obtained, extending a former result of Thomassen

    Interval matroids and graphs

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    AbstractA base of the cycle space of a binary matroid M on E is said to be convex if its elements can be totally ordered in such a way that for every e Īµ E the set of elements of the base containing e is an interval. We show that a binary matroid is cographic iff it has a convex base of cycles; equivalently, graphic matroids can be represented as ā€œinterval matroidsā€ (matroids associated in a natural way to interval systems). As a consequence, we obtain characterizations of planar graphs and cubic cyclically-4-edge-connected planar graphs in terms of convex bases of cycles
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