1,714 research outputs found
Entire choosability of near-outerplane graphs
It is proved that if G is a plane embedding of a K4-minor-free graph with maximum degree Î, then G is entirely 7-choosable if Îâ€4 and G is entirely (Î+ 2)-choosable if Îâ„ 5; that is, if every vertex, edge and face of G is given a list of max{7,Î+2} colours, then every element can be given a colour from its list such that no two adjacent or incident elements are given the same colour. It is proved also that this result holds if G is a plane embedding of a K2,3-minor-free graph or a (K2 + (K1 U K2))-minor-free graph. As a special case this proves that the Entire Colouring Conjecture, that a plane graph is entirely (Î + 4)-colourable, holds if G is a plane embedding of a K4-minor-free graph, a K2,3-minor-free graph or a (K2 + (K1 U K2))-minor-free graph
Every plane graph of maximum degree 8 has an edge-face 9-colouring
An edge-face colouring of a plane graph with edge set and face set is
a colouring of the elements of such that adjacent or incident
elements receive different colours. Borodin proved that every plane graph of
maximum degree can be edge-face coloured with colours.
Borodin's bound was recently extended to the case where . In this
paper, we extend it to the case .Comment: 29 pages, 1 figure; v2 corrects a contraction error in v1; to appear
in SIDM
Bipartite partial duals and circuits in medial graphs
It is well known that a plane graph is Eulerian if and only if its geometric
dual is bipartite. We extend this result to partial duals of plane graphs. We
then characterize all bipartite partial duals of a plane graph in terms of
oriented circuits in its medial graph.Comment: v2: minor changes. To appear in Combinatoric
Percolation on self-dual polygon configurations
Recently, Scullard and Ziff noticed that a broad class of planar percolation
models are self-dual under a simple condition that, in a parametrized version
of such a model, reduces to a single equation. They state that the solution of
the resulting equation gives the critical point. However, just as in the
classical case of bond percolation on the square lattice, self-duality is
simply the starting point: the mathematical difficulty is precisely showing
that self-duality implies criticality. Here we do so for a generalization of
the models considered by Scullard and Ziff. In these models, the states of the
bonds need not be independent; furthermore, increasing events need not be
positively correlated, so new techniques are needed in the analysis. The main
new ingredients are a generalization of Harris's Lemma to products of partially
ordered sets, and a new proof of a type of Russo-Seymour-Welsh Lemma with
minimal symmetry assumptions.Comment: Expanded; 73 pages, 24 figure
Dessins, their delta-matroids and partial duals
Given a map on a connected and closed orientable surface, the
delta-matroid of is a combinatorial object associated to which captures some topological information of the embedding. We explore how
delta-matroids associated to dessins d'enfants behave under the action of the
absolute Galois group. Twists of delta-matroids are considered as well; they
correspond to the recently introduced operation of partial duality of maps.
Furthermore, we prove that every map has a partial dual defined over its field
of moduli. A relationship between dessins, partial duals and tropical curves
arising from the cartography groups of dessins is observed as well.Comment: 34 pages, 20 figures. Accepted for publication in the SIGMAP14
Conference Proceeding
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