Partial duality and closed 2-cell embeddings

In 2009 Chmutov introduced the idea of partial duality for embeddings of graphs in surfaces. We discuss some alternative descriptions of partial duality, which demonstrate the symmetry between vertices and faces. One is in terms of band decompositions, and the other is in terms of the gem (graph-encoded map) representation of an embedding. We then use these to investigate when a partial dual is a closed 2-cell embedding, in which every face is bounded by a cycle in the graph. We obtain a necessary and sufficient condition for a partial dual to be closed 2-cell, and also a sufficient condition for no partial dual to be closed 2-cell.Comment: 16 pages, 6 figures, 1 table. To appear in "Journal of combinatorics

Orientable embeddings and orientable cycle double covers of projective-planar graphs

In a closed 2-cell embedding of a graph each face is homeomorphic to an open disk and is bounded by a cycle in the graph. The Orientable Strong Embedding Conjecture says that every 2-connected graph has a closed 2-cell embedding in some orientable surface. This implies both the Cycle Double Cover Conjecture and the Strong Embedding Conjecture. In this paper we prove that every 2-connected projective-planar cubic graph has a closed 2-cell embedding in some orientable surface. The three main ingredients of the proof are (1) a surgical method to convert nonorientable embeddings into orientable embeddings; (2) a reduction for 4-cycles for orientable closed 2-cell embeddings, or orientable cycle double covers, of cubic graphs; and (3) a structural result for projective-planar embeddings of cubic graphs. We deduce that every 2-edge-connected projective-planar graph (not necessarily cubic) has an orientable cycle double cover.Comment: 16 pages, 3 figure

The Chv\'atal-Erd\H{o}s condition for prism-Hamiltonicity

The prism over a graph $G$ is the cartesian product $G \Box K_2$. It is known that the property of having a Hamiltonian prism (prism-Hamiltonicity) is stronger than that of having a $2$-walk (spanning closed walk using every vertex at most twice) and weaker than that of having a Hamilton path. For a graph $G$, it is known that $\alpha(G) \leq 2 \kappa(G)$, where $\alpha(G)$ is the independence number and $\kappa(G)$ is the connectivity, imples existence of a $2$-walk in $G$, and the bound is sharp. West asked for a bound on $\alpha (G)$ in terms of $\kappa (G)$ guaranteeing prism-Hamiltonicity. In this paper we answer this question and prove that $\alpha(G) \leq 2 \kappa(G)$ implies the stronger condition, prism-Hamiltonicity of $G$.Comment: 7 pages, no figure

Criticality of counterexamples to toroidal edge-hamiltonicity

A well-known conjecture of Gr\"unbaum and Nash-Williams proposes that 4-connected toroidal graphs are hamiltonian. The corresponding results for 4-connected planar and projective-planar graphs were proved by Tutte and by Thomas and Yu, respectively, using induction arguments that proved a stronger result, that every edge is on a hamilton cycle. However, this stronger property does not hold for 4-connected toroidal graphs: Thomassen constructed counterexamples. Thus, the standard inductive approach will not work for the torus. One possible way to modify it is by characterizing the situations where some edge is not on a hamilton cycle. We provide a contribution in this direction, by showing that the obvious generalizations of Thomassen's counterexamples are critical in a certain sense.Comment: 8 page

One-way infinite 2-walks in planar graphs

We prove that every 3-connected 2-indivisible infinite planar graph has a 1-way infinite 2-walk. (A graph is 2-indivisible if deleting finitely many vertices leaves at most one infinite component, and a 2-walk is a spanning walk using every vertex at most twice.) This improves a result of Timar, which assumed local finiteness. Our proofs use Tutte subgraphs, and allow us to also provide other results when the graph is bipartite or an infinite analog of a triangulation: then the prism over the graph has a spanning 1-way infinite path.Comment: 23 pages, 4 figure

Edge-outer graph embedding and the complexity of the DNA reporter strand problem

In 2009, Jonoska, Seeman and Wu showed that every graph admits a route for a DNA reporter strand, that is, a closed walk covering every edge either once or twice, in opposite directions if twice, and passing through each vertex in a particular way. This corresponds to showing that every graph has an \emph{edge-outer embedding}, that is, an orientable embedding with some face that is incident with every edge. In the motivating application, the objective is such a closed walk of minimum length. Here we give a short algorithmic proof of the original existence result, and also prove that finding a shortest length solution is NP-hard, even for $3$-connected cubic ($3$-regular) planar graphs. Independent of the motivating application, this problem opens a new direction in the study of graph embeddings, and we suggest new problems emerging from it.Comment: 16 pages, 7 figures, minor revisio

Toughness and prism-hamiltonicity of P4P_4-free graphs

The \emph{prism} over a graph $G$ is the product $G \Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for $k \geq 1$ a $P_4$-free graph has a spanning \emph{$k$-walk} (closed walk using each vertex at most $k$ times) if and only if it is $\frac{1}{k}$-tough. As our main result, we show that for the class of $P_4$-free graphs, the three properties of being prism-hamiltonian, having a spanning $2$-walk, and being $\frac{1}{2}$-tough are all equivalent

Toughness and spanning trees in K4K_4-minor-free graphs

For an integer $k$, a $k$-tree is a tree with maximum degree at most $k$. More generally, if $f$ is an integer-valued function on vertices, an $f$-tree is a tree in which each vertex $v$ has degree at most $f(v)$. Let $c(G)$ denote the number of components of a graph $G$. We show that if $G$ is a connected $K_4$-minor-free graph and $$c(G-S) \;\le\; \sum_{v \in S} (f(v)-1) \quad\hbox{for all $S \subseteq V(G)$ with $S \ne \emptyset$}$$ then $G$ has a spanning $f$-tree. Consequently, if $G$ is a $\frac{1}{k-1}$-tough $K_4$-minor-free graph, then $G$ has a spanning $k$-tree. These results are stronger than results for general graphs due to Win (for $k$-trees) and Ellingham, Nam and Voss (for $f$-trees). The $K_4$-minor-free graphs form a subclass of planar graphs, and are identical to graphs of treewidth at most $2$, and also to graphs whose blocks are series-parallel. We provide examples to show that the inequality above cannot be relaxed by adding $1$ to the right-hand side, and also to show that our result does not hold for general planar graphs. Our proof uses a technique where we incorporate toughness-related information into weights associated with vertices and cutsets.Comment: Proposition 2.3 in v1 was incorrect; this has been fixed. 25 pages, 1 figur

Minimal quadrangulations of surfaces

A quadrangular embedding of a graph in a surface $\Sigma$, also known as a quadrangulation of $\Sigma$, is a cellular embedding in which every face is bounded by a $4$-cycle. A quadrangulation of $\Sigma$ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in $\Sigma$. In this paper we determine $n(\Sigma)$, the order of a minimal quadrangulation of a surface $\Sigma$, for all surfaces, both orientable and nonorientable. Letting $S_0$ denote the sphere and $N_2$ the Klein bottle, we prove that $n(S_0)=4, n(N_2)=6$, and $n(\Sigma)=\lceil (5+\sqrt{25-16\chi(\Sigma)})/2\rceil$ for all other surfaces $\Sigma$, where $\chi(\Sigma)$ is the Euler characteristic. Our proofs use a `diagonal technique', introduced by Hartsfield in 1994. We explain the general features of this method.Comment: 25 pages, 20 figure

Quadrangular embeddings of complete graphs and the Even Map Color Theorem

Hartsfield and Ringel constructed orientable quadrangular embeddings of the complete graph $K_n$ for $n\equiv 5 \pmod 8$, and nonorientable ones for $n \ge 9$ and $n\equiv 1 \pmod 4$. These provide minimal quadrangulations of their underlying surfaces. We extend these results to determine, for every complete graph $K_n$, $n \ge 4$, the minimum genus, both orientable and nonorientable, for the surface in which $K_n$ has an embedding with all faces of degree at least $4$, and also for the surface in which $K_n$ has an embedding with all faces of even degree. These last embeddings provide sharpness examples for a result of Hutchinson bounding the chromatic number of graphs embedded with all faces of even degree, completing the proof of the Even Map Color Theorem. We also show that if a connected simple graph $G$ has a perfect matching and a cycle then the lexicographic product $G[K_4]$ has orientable and nonorientable quadrangular embeddings; this provides new examples of minimal quadrangulations.Comment: Version 2 is a greatly expanded version of the paper with four new authors (Lawrencenko, Chen, Hartsfield, Yang). It includes results on quadrangular or nearly quadrangular embeddings for all complete graphs. It also includes an application to a coloring result. This version contains some details omitted from the version that will be submitted for publication. 26 pages; 6 figure