Related papers: Checkerboard colourable twisted duals
The geometric dual of a cellularly embedded graph is a fundamental concept in graph theory and also appears in many other branches of mathematics. The partial dual is an essential generalization which can be obtained by forming the…
In this paper, we first introduce the notions of checkerboard colourable minors for ribbon graphs motivated by the Eulerian ribbon graph minors, and two kinds of bipartite minors for ribbon graphs, one of which is the dual of the…
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…
Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the…
Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of all-crossing directions of its medial graph. Then Metsidik and Jin characterized all Eulerian partial duals of a plane graph in terms of…
We consider two operations on an edge of an embedded graph (or equivalently a ribbon graph): giving a half-twist to the edge and taking the partial dual with respect to the edge. These two operations give rise to an action of S_3^{|E(G)|},…
The mutually enriching relationship between graphs and matroids has motivated discoveries in both fields. In this paper, we exploit the similar relationship between embedded graphs and delta-matroids. There are well-known connections…
We define pure graphs, invertible graphs, and the notion of complementation of bicoloured graphs. The study of pure graphs is motivated by two conjectures about the transition systems of eulerian graphs and by the Cycle Double Cover…
A colouring of a graph is "nonrepetitive" if for every path of even order, the sequence of colours on the first half of the path is different from the sequence of colours on the second half. We show that planar graphs have nonrepetitive…
A 2-edge-coloured graph $G$ is {\bf supereulerian} if $G$ contains a spanning closed trail in which the edges alternate in colours. An {\bf eulerian factor} of a 2-edge-coloured graph is a collection of vertex disjoint induced subgraphs…
We generalize the natural duality of graphs embedded into a surface to a duality with respect to a subset of edges. The dual graph might be embedded into a different surface. We prove a relation between the signed Bollobas-Riordan…
We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial…
Gross, Mansour and Tucker introduced the partial-dual polynomial of a ribbon graph and asked under what conditions such a polynomial is even-interpolating, odd-interpolating, or both. In this paper, we provide an answer to this open…
Elementary graphs are graphs whose edges can be colored using two colors in such a way that the edges in any induced $P_3$ get distinct colors. They constitute a subclass of the class of claw-free perfect graphs. In this paper, we show that…
Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They showed that the partial-dual genus polynomial for an orientable ribbon graph is interpolating and gave an…
Recently, Gross, Mansour and Tucker introduced the partial-twuality polynomials. In this paper, we find that when there are enough parallel edges, any multiple graph is a negative answer to the problem 8.7 in their paper [European J.…
A \emph{directional labeling} of an edge $\emph{uv}$ in a graph $G=(V,E)$ by an ordered pair $ab$ is a labeling of the edge $uv$ such that the label on $uv$ in the direction from $u$ to $v$ is $\ell(uv)=ab$, and $\ell(vu)=ba$. New…
This work studies certain aspects of graphs embedded on surfaces. Initially, a colored graph model for a map of a graph on a surface is developed. Then, a concept analogous to (and extending) planar graph is introduced in the same spirit as…
We develop an algebraic framework for ribbon graphs, revealing symmetry properties of (partial) twisted duality. The original ribbon group action of Ellis-Monaghan and Moffatt restricts self-duality, -petriality, or -triality to the…
Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research…