Related papers: Strong embeddings of minimum genus
Simplicial surfaces describe the incidence relations between vertices, edges and faces of triangulated 2-dimensional manifolds in a purely combinatorial way. By considering only the incidences of edges and faces, simplicial surfaces are…
This article focuses on a class of properly edge-colored graphs, which arise from topological combinatorics, and investigates their embeddings onto surfaces. Specifically, these graphs are known as the dual graphs of balanced normal…
We show that, for every n and every surface $\Sigma$, there is a graph U embeddable on $\Sigma$ with at most cn^2 vertices that contains as minor every graph embeddable on $\Sigma$ with n vertices. The constant c depends polynomially on the…
A {\em rooted graph} is a graph together with a designated vertex subset, called the {\em roots}. In this paper, we consider rooted graphs embedded in a fixed surface. A collection of faces of the embedding is a {\em face cover} if every…
In 1930, Ramsey proved that every large graph contains either a large clique or a large edgeless graph as an induced subgraph. It is well known that every large connected graph contains a long path, a large clique, or a large star as an…
Given a bridgeless graph $G$, the Cycle Double Cover Conjecture posits that there is a list of cycles of $G$, such that every edge appears in exactly two cycles on the list. This conjecture was originally posed independently in 1973 by…
The Union Closed Sets Conjecture is one of the most renowned problems in combinatorics. Its appeal lies in the simplicity of its statement contrasted with the potential complexity of its resolution. The conjecture posits that, in any union…
A graph is nearly embedded in a surface if it consists of graph $G_0$ that is embedded in the surface, together with a bounded number of vortices having no large transactions. It is shown that every large wall (or grid minor) in a nearly…
Many conjectures and open problems in graph theory can either be reduced to cubic graphs or are directly stated for cubic graphs. Furthermore, it is known that for a lot of problems, a counterexample must be a snark, i.e. a bridgeless cubic…
Given a bridgeless graph G, the well-known cycle double cover conjecture posits that there is a list of cycles of G, such that every edge appears in exactly two cycles. In this paper, we prove the cycle double cover conjecture. More…
A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except $K_8$, the complete graph…
Random 2-cell embeddings of a given graph $G$ are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, $\mathbb{E}[F_G]$, of such an embedding which is equivalent to studying its average…
This is a companion paper to the paper "Hyperstability in the Erdos-Sos Conjecture". In that paper the following rough structure theorem was proved for graphs G containing no copy of a bounded degree tree T: from any such G, one can delete…
For a graph $G$, and a nonnegative integer $g$, let $a_g(G)$ be the number of $2$-cell embeddings of $G$ in an orientable surface of genus $g$ (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker…
Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of…
An ordinary voltage graph embedding of a graph in a surface encodes a certain kind of highly symmetric covering space of that surface. Given an ordinary voltage graph embedding of a graph $G$ in a surface with voltage group $A$ and a…
Recently, the bunkbed conjecture has been shown to be false, which naturally prompts questions on how to classify the graphs that still satisfy the conjecture. We distinguish between a weak version of the bunkbed conjecture where all the…
The cycle double cover conjecture states that a graph is bridge-free if and only if there is a family of edge-simple cycles such that each edge is contained in exactly two of them. It was formulated independently by Szekeres (1973) and…
Tutte's embedding theorem states that every 3-connected graph without a $K_5$ or $K_{3,3}$ minor (i.e. a planar graph) is embedded in the plane if the outer face is in convex position and the interior vertices are convex combinations of…
It is known that graphs cellularly embedded into surfaces are equivalent to ribbon graphs. In this work, we generalize this statement to broader classes of graphs and surfaces. Half-edge graphs extend abstract graphs and are useful in…