Obstructions for two-vertex alternating embeddings of graphs in surfaces (original) (raw)

An obstruction to embedding graphs in surfaces

Discrete Mathematics, 1989

It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.

Embedding Digraphs on Orientable Surfaces

Journal of Combinatorial Theory, Series B, 2002

We consider a notion of embedding digraphs on orientable surfaces, applicable to digraphs in which the indegree equals the outdegree for every vertex, i.e., Eulerian digraphs. This idea has been considered before in the context of "compatible Euler tours" or "orthogonal A-trails" by Andsersen at al [1] and by Bouchet [4]. This prior work has mostly been limited to embeddings of Eulerian digraphs on predetermined surfaces, and to digraphs with underlying graphs of maximum degree at most 4. In this paper, a foundation is laid for the study of all Eulerian digraph embeddings. Results are proved which are analogous to those fundamental to the theory of undirected graph embeddings, such as Duke's Theorem [5], and an infinite family of digraphs which demonstrates that the genus range for an embeddable digraph can be any nonnegative integer is given. We show that it is possible to have genus range equal to one, with arbitrarily large minimum genus, unlike in the undirected case. The difference between the minimum genera of a digraph and its underlying graph is considered, as is the difference between the maximum genera. We say that a digraph is upper-embeddable if it can be embedded with 2 or 3 regions, and prove that every regular tournament is upper-embeddable.

Regular orientable imbeddings of complete graphs

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.

Flexibility of Polyhedral Embeddings of Graphs in Surfaces

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.

Closed 2-cell embeddings of graphs with no -minors

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.

Graph minors and graphs on surfaces

Surveys in Combinatorics, 2001, 2001

Graph minors and the theory of graphs embedded in surfaces are fundamentally interconnected. Robertson and Seymour used graph minors to prove a generalization of the Kuratowski Theorem to arbitrary surfaces [37], while they also need surface embeddings in their Excluded Minor Theorem [45]. Various recent results related to graph minors and graphs on surfaces are presented.

Strong embeddings of minimum genus

Discrete Mathematics, 2010

A "folklore conjecture, probably due to Tutte" (as described in [P.D. Seymour, Sums of circuits, Graph theory and related topics (Proc. Conf., Univ. Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph. Sporadic counterexamples to this conjecture have been known since the late 1970's. In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional to the order of these graphs), thus providing a plethora of strong counterexamples to the above conjecture. The main result yielding such counterexamples may be of independent interest.

Circular embeddings of planar graphs in nonspherical surfaces

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.