Eulerian partial duals of plane graphs

In the paper Combinatorica 33(2) (2013) 231–252, Huggett and Moffatt characterized all bipartite partial duals of a plane graph in terms of oriented circuits in its medial graph. An open problem posed in their paper is the characterization of Eulerian partial duals of plane graphs. In this article, we solve this problem by considering half‐edge orientations of medial graphs.     

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.     

Excluded checkerboard colourable ribbon graph minors

Motivated by the Eulerian ribbon graph minors, in this paper we introduce the notion of checkerboard colourable minors for ribbon graphs and its dual: bipartite minors for ribbon graphs. Motivated by the bipartite minors of abstract graphs, another bipartite minors for ribbon graphs, i.e. the bipartite ribbon graph join minors are also introduced. Using these minors then we give excluded minor characterizations of the classes of checkerboard colourable ribbon graphs, bipartite ribbon graphs, plane checkerboard colourable ribbon graphs and plane bipartite ribbon graphs.     

Eulerian and even-face ribbon graph minors

In this paper, we introduce Eulerian and even-face ribbon graph minors. These minors preserve Eulerian and even-face properties of ribbon graphs, respectively. We then characterize Eulerian, even-face, plane Eulerian and plane even-face ribbon graphs using these minors.     

On a conjecture of Gross,Mansour and Tucker

Partial duality is a duality of ribbon graphs relative to a subset of their edges generalizing the classical Euler–Poincaré duality. This operation often changes the genus. Recently J.L. Gross, T. Mansour, and T.W. Tucker formulated a conjecture that for any ribbon graph different from plane trees and their partial duals, there is a subset of edges partial duality relative to which does change the genus. A family of counterexamples was found by Qi Yan and Xian’an Jin. In this note we prove that essentially these are the only counterexamples.     

A Penrose polynomial for embedded graphs

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 which cannot be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion–contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable chequerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G$G$ can be expressed as a sum of chromatic polynomials of twisted duals of G$G$. This allows us to obtain a new reformulation of the Four Colour Theorem.     

The uniquely embeddable planar graphs

Whitney [7] proved in 1932 that for any two embeddings of a planar 3-connected graph, their combinatorial duals are isomorphic. In this manner, the term “uniquely embeddable planar graph” was introduced. It is a well-known fact that combinatorial and geometrical duals are equivalent concepts. In this paper, the concept of unique embeddability is introduced in terms of special types of isomorphisms between any two embeddings of a planar graph. From this, the class U of all graphs which are uniquely embeddable in the plane according to this definition, is determined, and the planar 3-connected graphs are a proper subset of U. It turns out that the graphs in U have a unique geometrical dual (i.e., for any two embeddings of such a graph, their geometrical duals are isomorphic). Furthermore, the theorems and their proofs do not involve any type of duals.     

Excluded Minors and the Ribbon Graphs of Knots

In this article we consider minors of ribbon graphs (or, equivalently, cellularly embedded graphs). The theory of minors of ribbon graphs differs from that of graphs in that contracting loops is necessary and doing this can create additional vertices and components. Thus, the ribbon graph minor relation is incompatible with the graph minor relation. We discuss excluded minor characterizations of minor closed families of ribbon graphs. Our main result is an excluded minor characterization of the family of ribbon graphs that represent knot and link diagrams.     

Rectangular-radial drawings of cubic plane graphs

Rectangular drawings and rectangular duals can be naturally extended to other surfaces. In this paper, we extend rectangular drawings and rectangular duals to drawings on a cylinder. The extended drawings are called rectangular-radial drawings and rectangular-radial duals. Rectangular-radial drawings correspond to periodic rectangular tilings of a 1-dimensional strip. We establish a necessary and sufficient condition for plane graphs with maximum degree 3 to have rectangular-radial drawings and a necessary and sufficient condition for triangulated plane graphs to have rectangular-radial duals. Furthermore, we present three linear time algorithms under three different conditions for finding a rectangular-radial drawing for a given cubic plane graph, if one exists.     

The traveling salesman problem in graphs with some excluded minors

Given a graph and a length function defined on its edge-set, the Traveling Salesman Problem can be described as the problem of finding a family of edges (an edge may be chosen several times) which forms a spanning Eulerian subgraph of minimum length. In this paper we characterize those graphs for which the convex hull of all solutions is given by the nonnegativity constraints and the classical cut constraints. This characterization is given in terms of excluded minors. A constructive characterization is also given which uses a small number of basic graphs.     

Hamiltonicity in vertex envelopes of plane cubic graphs

In this paper we study a graph operation which produces what we call the “vertex envelope” G^∗_V from a graph G. We apply it to plane cubic graphs and investigate the hamiltonicity of the resulting graphs, which are also cubic. To this end, we prove a result giving a necessary and sufficient condition for the existence of hamiltonian cycles in the vertex envelopes of plane cubic graphs. We then use these conditions to identify graphs or classes of graphs whose vertex envelopes are either all hamiltonian or all non-hamiltonian, paying special attention to bipartite graphs. We also show that deciding if a vertex envelope is hamiltonian is NP-complete, and we provide a polynomial algorithm for deciding if a given cubic plane graph is a vertex envelope.     

Partial duality and Bollobás and Riordan’s ribbon graph polynomial

Recently S. Chmutov introduced a generalization of the dual of a ribbon graph (or equivalently an embedded graph) and proved a relation between Bollobás and Riordan’s ribbon graph polynomial of a ribbon graph and of its generalized duals. Here I show that the duality relation satisfied by the ribbon graph polynomial can be understood in terms of knot theory and I give a simple proof of the relation which used the homfly polynomial of a knot.     

Non-zero component union graph of a finite-dimensional vector space

In this paper, we introduce a graph structure, called non-zero component union graph on finite-dimensional vector spaces. We show that the graph is connected and find its domination number, clique number and chromatic number. It is shown that two non-zero component union graphs are isomorphic if and only if the base vector spaces are isomorphic. In case of finite fields, we study the edge-connectivity and condition under which the graph is Eulerian. Moreover, we provide a lower bound for the independence number of the graph. Finally, we come up with a structural characterization of non-zero component union graph.     

Note on 3-choosability of planar graphs with maximum degree 4

Deciding whether a planar graph (even of maximum degree 4) is 3-colorable is NP-complete. Determining subclasses of planar graphs being 3-colorable has a long history, but since Grötzsch’s result that triangle-free planar graphs are such, most of the effort was focused to solving Havel’s and Steinberg’s conjectures. In this paper, we prove that every planar graph obtained as a subgraph of the medial graph of any bipartite plane graph is 3-choosable. These graphs are allowed to have close triangles (even incident), and have no short cycles forbidden, hence representing an entirely different class than the graphs inferred by the above mentioned conjectures.     

基圈数为5的2-连通平图的构造

所有的2-连通平图可通过收缩2度点变换成无2度点的、基圈数不变的2-连通平图.本文给出了基圈数为5的、无2度点的所有2-连通平图.     

Planar unit-distance graphs having planar unit-distance complement

A graph G=(V,E) is called a unit-distance graph in the plane if there is an embedding of V into the plane such that every pair of adjacent vertices are at unit distance apart. If an embedding of V satisfies the condition that two vertices are adjacent if and only if they are at unit distance apart, then G is called a strict unit-distance graph in the plane. A graph G is a (strict) co-unit-distance graph, if both G and its complement are (strict) unit-distance graphs in the plane. We show by an exhaustive enumeration that there are exactly 69 co-unit-distance graphs (65 are strict co-unit-distance graphs), 55 of which are connected (51 are connected strict co-unit-distance graphs), and seven are self-complementary.     

Series parallel extensions of plane graphs to dual-eulerian graphs

A plane graph is dual-eulerian if it has an eulerian tour with the property that the same sequence of edges also forms an eulerian tour in the dual graph. Dual-eulerian graphs are of interest in the design of CMOS VLSI circuits.Every dual-eulerian plane graph also has an eulerian Petrie (left–right) tour thus we consider series-parallel extensions of plane graphs to graphs, which have eulerian Petrie tours. We reduce several special cases of extensions to the problem of finding hamiltonian cycles. In particular, a 2-connected plane graph G has a single series parallel extension to a graph with an eulerian Petrie tour if and only if its medial graph has a hamiltonian cycle.     

On Spin Models,Triply Regular Association Schemes,and Duality

Motivated by the construction of invariants of links in 3-space, we study spin models on graphs for which all edge weights (considered as matrices) belong to the Bose-Mesner algebra of some association scheme. We show that for series-parallel graphs the computation of the partition function can be performed by using series-parallel reductions of the graph appropriately coupled with operations in the Bose-Mesner algebra. Then we extend this approach to all plane graphs by introducing star-triangle transformations and restricting our attention to a special class of Bose-Mesner algebras which we call exactly triply regular. We also introduce the following two properties for Bose-Mesner algebras. The planar duality property (defined in the self-dual case) expresses the partition function for any plane graph in terms of the partition function for its dual graph, and the planar reversibility property asserts that the partition function for any plane graph is equal to the partition function for the oppositely oriented graph. Both properties hold for any Bose-Mesner algebra if one considers only series-parallel graphs instead of arbitrary plane graphs. We relate these notions to spin models for link invariants, and among other results we show that the Abelian group Bose-Mesner algebras have the planar duality property and that for self-dual Bose-Mesner algebras, planar duality implies planar reversibility. We also prove that for exactly triply regular Bose-Mesner algebras, to check one of the above properties it is sufficient to check it on the complete graph on four vertices. A number of applications, examples and open problems are discussed.     

Sufficient conditions for a graph to be Hamiltonian

Three sufficient conditions for a graph to be Hamiltonian are given. These theorems are in terms of subgraph structure and do not require the fairly high global line density which is basic to the Pósa-like sufficiency conditions. Line graphs of both Eulerian graphs and Hamiltonian graphs are also characterized.     

图论中若干基本概念的历史注记

考察了图与子图,树,匹配,欧拉图与哈密尔顿图,可平面图,以及与图的连通性和图的着色有关的若干图论基本概念的历史背景.