Related papers: Whitney's Theorem for 2-Regular Planar Digraphs

Whitney's 2-Isomorphism Theorem characterises when two graphs have isomorphic cycle matroids. We present an analogue of this theorem for graphs embedded in surfaces by characterising when two graphs in surface have isomorphic…

We prove that for any parameter r an r-locally 2-connected graph G embeds r-locally planarly in a surface if and only if a certain matroid associated to the graph G is co-graphic. This extends Whitney's abstract planar duality theorem from…

A well-known theorem of Whitney states that a 3-connected planar graph admits an essentially unique embedding into the 2-sphere. We prove a 3-dimensional analogue: a simply-connected $2$-complex every link graph of which is 3-connected…

We give a non-abelian analogue of Whitney's 2-isomorphism theorem for graphs. Whitney's theorem states that the cycle space determines a graph up to 2-isomorphism. Instead of considering the cycle space of a graph which is an abelian…

A graph $G$ is a link-irregular graph if every two distinct vertices of $G$ have non-isomorphic links. The link of a vertex $v$ in $G$ is the subgraph induced by the neighbors of $v$ in $G$. Ali, Chartrand and Zhang [Discussiones…

In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring…

An embedding of a graph on an orientable surface is orientably-regular (or rotary, in an equivalent terminology) if the group of orientation-preserving automorphisms of the embedding is transitive (and hence regular) on incident vertex-edge…

In this note we give a short and elementary proof of a more general version of Whitney's theorem that 3-connected planar graphs have a unique embedding in the plane. A consequence of the theorem is that cubic plane graphs cannot be embedded…

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…

Whitney's theorem states that every 3-connected planar graph is uniquely embeddable on the sphere. On the other hand, it has many inequivalent embeddings on another surface. We shall characterize structures of a $3$-connected $3$-regular…

In a planar L-drawing of a directed graph (digraph) each edge e is represented as a polyline composed of a vertical segment starting at the tail of e and a horizontal segment ending at the head of e. Distinct edges may overlap, but not…

A 2-cell embedding of a graph $G$ into a closed (orientable or nonorientable) surface is called regular if its automorphism group acts regularly on the flags - mutually incident vertex-edge-face triples. In this paper, we classify the…

In this paper, we study the stability result of a well-known theorem of Bondy. We prove that for any 2-connected non-hamiltonian graph, if every vertex except for at most one vertex has degree at least $k$, then it contains a cycle of…

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…

Tutte showed that a graph $G$ is planar if and only if the conflict graph associated to every cycle of $G$ is bipartite. We define a (not necessarily unique) signed conflict graph associated to a maximally planar subgraph of a nonplanar…

A tuple (s1,t1,s2,t2) of vertices in a simple undirected graph is 2-linked when there are two vertex-disjoint paths respectively from s1 to t1 and s2 to t2. A graph is 2-linked when all such tuples are 2-linked. We give a new and simple…

A digraph $H$ is called ubiquitous if every digraph that contains arbitrarily many vertex-disjoint copies of $H$ also contains infinitely many vertex-disjoint copies of $H$. We study oriented double rays, that is, digraphs $H$ whose…

In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in $2$-connected or $2$-edge connected graphs. We provide some similar conditions for the existence of an…

It is shown that the number of pages required for a book embedding of a graph is the maximum of the numbers needed for any of the maximal nonseparable subgraphs and that a plane graph in which every triangle bounds a face has a two-page…

An embedding of a graph in a book, called book embedding, consists of a linear ordering of its vertices along the spine of the book and an assignment of its edges to the pages of the book, so that no two edges on the same page cross. The…