79 research outputs found
Embedding Digraphs on Orientable Surfaces
AbstractWe 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 Andersen and by Bouchet. 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 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 two or three regions and prove that every regular tournament is upper-embeddable
The enumeration of planar graphs via Wick's theorem
A seminal technique of theoretical physics called Wick's theorem interprets
the Gaussian matrix integral of the products of the trace of powers of
Hermitian matrices as the number of labelled maps with a given degree sequence,
sorted by their Euler characteristics. This leads to the map enumeration
results analogous to those obtained by combinatorial methods. In this paper we
show that the enumeration of the graphs embeddable on a given 2-dimensional
surface (a main research topic of contemporary enumerative combinatorics) can
also be formulated as the Gaussian matrix integral of an ice-type partition
function. Some of the most puzzling conjectures of discrete mathematics are
related to the notion of the cycle double cover. We express the number of the
graphs with a fixed directed cycle double cover as the Gaussian matrix integral
of an Ihara-Selberg-type function.Comment: 23 pages, 2 figure
Interlacement in 4-regular graphs: a new approach using nonsymmetric matrices
Let be a 4-regular graph with an Euler system . We introduce a simple way to modify the interlacement matrix of so that every circuit partition of has an associated modified interlacement matrix . If  and are Euler systems of then and are inverses, and for any circuit partition , . This machinery allows for short proofs of several results regarding the linear algebra of interlacement
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