Orientable embeddings and orientable cycle double covers

Discrete Mathematics, 2001

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classiÿcation of internally-4-connected graphs with no V8-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.

Discrete Mathematics, 1994

We show that every 3-connected planar graph has a circular embedding in some nonspherical surface. More generally, we characterize those planar graphs that have a 2-representative embedding in some nonspherical surface.

Discrete Mathematics, 2001

A closed 2-cell embedding of a graph embedded in some surface is an embedding such that each face is bounded by a cycle in the graph. The strong embedding conjecture says that every 2-connected graph has a closed 2-cell embedding in some surface. In this paper, we prove that any 2-connected graph without V8 (the M obius 4-ladder) as a minor has a closed 2-cell embedding in some surface. As a corollary, such a graph has a cycle double cover. The proof uses a classiÿcation of internally-4-connected graphs with no V8-minor (due to Kelmans and independently Robertson), and the proof depends heavily on such a characterization.

Canadian Journal of Mathematics, 1992

Let G be a graph embedded in a closed surface. The embedding is “locally planar” if for each face, a “large” neighbourhood of this face is simply connected. This notion is formalized, following [RV], by introducing the width ρ(ψ) of the embedding ψ. It is shown that embeddings with ρ(ψ) ≥ 3 behave very much like the embeddings of planar graphs in the 2-sphere. Another notion, “combinatorial local planarity”, is introduced. The criterion is independent of embeddings of the graph, but it guarantees that a given cycle in a graph G must be contractible in any minimal genus embedding of G (either orientable, or non-orientable). It generalizes the width introduced before. As application, short proofs of some important recently discovered results about embeddings of graphs are given and generalized or improved. Uniqueness and switching equivalence of graphs embedded in a fixed surface are also considered.

Journal of Combinatorial Theory, Series B, 2001

Whitney's theorem states that 3-connected planar graphs admit essentially unique embeddings in the plane. We generalize this result to embeddings of graphs in arbitrary surfaces by showing that there is a function ξ : N 0 → N 0 such that every 3-connected graph admits at most ξ(g) combinatorially distinct embeddings of face-width ≥ 3 into surfaces whose Euler genus is at most g.

Combinatorics, Probability and Computing, 2006

Polyhedral embeddings of cubic graphs by means of certain operations are studied. It is proved that some known families of snarks have no (orientable) polyhedral embeddings. This result supports a conjecture of Grünbaum that no snark admits an orientable polyhedral embedding. This conjecture is verified for all snarks having up to 30 vertices using computer. On the other hand, for every nonorientable surface S, there exists a non 3-edge-colorable graph, which polyhedrally embeds in S.

Discrete Mathematics, 1996

It is shown that embeddings of planar graphs in the projective plane have very specific structure. By exhibiting this structure we indirectly characterize graphs on the projective plane whose dual graphs are planar. Whitney's Theorem about 2-switching equivalence of planar embeddings is generalized: Any two embeddings of a planar graph in the projective plane can be obtained from each other by means of simple local reembeddings, very similar to Whitney's switchings.

European Journal of Combinatorics, 2017

A class of graphs that lies strictly between the classes of graphs of genus (at most) k − 1 and k is studied. For a fixed orientable surface S k of genus k, let A k xy be the minor-closed class of graphs with terminals x and y that either embed into S k−1 or admit an embedding Π into S k such that there is a Π-face where x and y appear twice in the alternating order. In this paper, the obstructions for the classes A k xy are studied. In particular, the complete list of obstructions for A 1 xy is presented.

Journal of Combinatorial Theory, Series B, 1985

This paper classifies the regular imbeddings of the complete graphs K,, in orientable surfaces. Biggs showed that these exist if and only if n is a prime power p', his examples being Cayley maps based on the finite field F= GF(n). We show that these are the only examples, and that there are q5(n-1)/e isomorphism classes of such maps (where 4 is Euler's function), each corresponding to a conjugacy class of primitive elements of F, or equivalently to an irreducible factor of the cyclotomic polynomial Qn-r(z) over GF(p). We show that these maps are all equivalent under Wilson's map-operations Hi, and we determined for which n they are reflexible or self-dual.

European Journal of Combinatorics, 1997

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i. e. an embedding that each face is bounded by a circuit in the graph. A graph is called k-crosscap embeddable if it can be embedded in the surface of non-orientable genus k. We confirm the strong embedding conjecture for 5-crosscap embed-461