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Checkerboard Colourable Twuals
In this article we show that any embedded graph has a checkerboard colourable twual, which is equivalent to having a bipartite twual. We also obtain...
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Eulerian and Bipartite Binary Delta-matroids
Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is...
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The Number of Circles of a Maximum State of a Plane Graph with Applications
Motivated by the connection with the genus of the corresponding link and its application on DNA polyhedral links, in this paper, we introduce a...
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Counterexample to an Extension of the Hanani-Tutte Theorem on the Surface of Genus 4
We find a graph of genus 5 and its drawing on the orientable surface of genus 4 with every pair of independent edges crossing an even number of...
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Once Punctured Disks, Non-Convex Polygons, and Pointihedra
We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of...
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Fundamental groupoids of digraphs and graphs
We introduce the notion of fundamental groupoid of a digraph and prove its basic properties. In particular, we obtain a product theorem and an...
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4-colored Graphs and Knot/Link Complements
A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. 4-regular graphs endowed with a proper...
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Surgeries of pairing of Edges associated to trivalent graphs
Our purpose is to study the pairings of edges of hyperbolic polygonswith p edges, who are associated with a tessellation { p , q } = {12 g −6, 3}, where g ...
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Constructions of Binary Codes Based on Bipartite Graphs
In this paper, we construct some families of binary linear and nonlinear codes by using various bipartite graphs. Furthermore, we also determine some...
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Embedding a graph-like continuum in a surface
Graph-like continua provide a very natural setting for generalizing finite graphs to infinite, compact structures. For example, the Freudenthal...
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Irreducible Triangulations of Surfaces with Boundary
A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we...
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Critical Groups of Simplicial Complexes
We generalize the theory of critical groups from graphs to simplicial complexes. Specifically, given a simplicial complex, we define a family of...
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Notes on tiled incompressible tori
Let Θ denote the class of essential tori in a closed braid complement which admit a standard tiling in the sense of Birman and Menasco [Birman J.S.,...
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Homogeneous links, Seifert surfaces, digraphs and the reduced Alexander polynomial
We give a geometric proof of the following result of Juhasz. Let a g be the leading coefficient of the Alexander polynomial of an alternating knot K....
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Level Eulerian Posets
The notion of level posets is introduced. This class of infinite posets has the property that between every two adjacent ranks the same bipartite...
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Complementary Regions of Knot and Link Diagrams
An increasing sequence of integers is said to be universal for knots and links if every knot and link has a reduced projection on the sphere such...
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A local criterion for Tverberg graphs
The topological Tverberg theorem states that for any prime power q and continuous map from a ( d +1)( q −1)-simplex to ℝ d , there are q disjoint faces F ...
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On planarity of compact, locally connected, metric spaces
Independently, Claytor [Ann. Math. 35 (1934), 809–835] and Thomassen [Combinatorica 24 (2004), 699–718] proved that a 2-connected, compact, locally...