Growth rate of canonical and minimal group embeddings of spherical latin trades

Grubman, Tony; Wanless, Ian M.

doi:10.1016/j.jcta.2013.11.006

Cited by 4 publications

(7 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Batagelj [1] introduced two operations on plane Eulerian triangulations by means of which larger such maps can be generated from smaller ones. These operations have been used and extended in several papers (via the aforementioned correspondences) to generate latin bitrades [14,24]. The inverse of one of these operations, translated to alternating dimaps, corresponds to (technically, restricted versions of) our minor operations.…”

Section: History and Related Workmentioning

confidence: 99%

“…We can calculate F (L 2,1 + e ω 2 ) by applying (21) at f (or (23) at e), obtaining xzw. Alternatively, we can apply (24) at g, obtaining (x + z + w)w. Equating the results and using w = 0, we obtain…”

Section: Definitionmentioning

confidence: 99%

“…A Tutte invariant for alternating dimaps is defined as for a simple Tutte invariant, except that condition 5, with (24), is replaced by a requirement that there exist nonzero a, b, c such that, for any alternating dimap G and any e ∈ E(G),…”

Section: Definitionmentioning

confidence: 99%

See 2 more Smart Citations

Minors for alternating dimaps

Farr

2017

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

We develop a theory of minors for alternating dimaps -orientably embedded digraphs where, at each vertex, the incident edges (taken in the order given by the embedding) are directed alternately into, and out of, the vertex. We show that they are related by the triality relation of Tutte. They do not commute in general, though do in many circumstances, and we characterise the situations where they do. The relationship with triality is reminiscent of similar relationships for binary functions, due to the author, so we characterise those alternating dimaps which correspond to binary functions. We give a characterisation of alternating dimaps of at most a given genus, using a finite set of excluded minors. We also use the minor operations to define simple Tutte invariants for alternating dimaps and characterise them. We establish a connection with the Tutte polynomial, and pose the problem of characterising universal Tutte-like invariants for alternating dimaps based on these minor operations.

show abstract

Section: History and Related Workmentioning

confidence: 99%

Section: Definitionmentioning

confidence: 99%

See 1 more Smart Citation

Minors for alternating dimaps

Farr

2017

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…The group C in Theorem 1.3 is referred to as the canonical group of the spherical latin bitrade (see [14,18]).…”

Section: Embeddings Of Latin Bitrades Into Abelian Groupsmentioning

confidence: 99%

On which groups can arise as the canonical group of a spherical latin bitrade

Bonetta-Martin

McCourt

2017

Ars Math. Contemp.

View full text Add to dashboard Cite

We address a question of Cavenagh and Wanless asking: which finite abelian groups arise as the canonical group of a spherical latin bitrade? We prove the existence of an infinite family of finite abelian groups that do not arise as canonical groups of spherical latin bitrades. Using a connection between abelian sandpile groups of digraphs underlying directed Eulerian spherical embeddings, we go on to provide several, general, families of finite abelian groups that do arise as canonical groups. These families include:• any abelian group in which each component of the Smith Normal Form has composite order;• any abelian group with Smith Normal Form Z n p ⊕ k i=1 Z pa i , where 1 ≤ k, 2 ≤ a 1 , a 2 , . . . , a k , p and n ≤ 1 + 2 k i=1 (a i − 1); and • with one exception and three potential exceptions any abelian group of rank two.

show abstract

“…where C is a finite abelian group. In the same paper the question of the growth rate of the maximal order of C, in the terminology established in [15] the canonical group of the face 2-coloured spherical triangulation, was raised. More precisely: [2]).…”

Section: Introductionmentioning

confidence: 99%

Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere

McCourt¹

2016

Electron. J. Combin.

View full text Add to dashboard Cite

Let G be a properly face 2-coloured (say black and white) piecewiselinear triangulation of the sphere with vertex set V . Consider the abelian group A W generated by the set V , with relations r + c + s = 0 for all white triangles with vertices r, c and s. The group A B can be defined similarly, using black triangles. These groups are related in the following mannerThe finite torsion subgroup C is referred to as the canonical group of the triangulation. Let m t be the maximal order of C over all properly face two-coloured spherical triangulations with t triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for lim sup t→∞ (m t ) 1/t .

show abstract

Growth rate of canonical and minimal group embeddings of spherical latin trades

Cited by 4 publications

References 9 publications

Minors for alternating dimaps

Minors for alternating dimaps

On which groups can arise as the canonical group of a spherical latin bitrade

Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere

Contact Info

Product

Resources

About