Graphs of given genus and arbitrarily large maximum genus

The maximum genus of a connected graph G is the maximum among the genera of all compact orientable 2-manifolds upon which G has 2-cell embeddings. In the theorems that follow the use of an edge-adding technique is combined with the well-known Edmonds' technique to produce the desired results. Planar graphs of arbitrarily large maximum genus are displayed in Theorem 1. Theorem 2 shows that the possibility for arbitrarily large difference between genus and maximum genus is not limited to planar graphs. In particular, we show that the wheel graph, the standard maximal planar graph, and the prism graph are upper embeddable. We then show that given m and n, there is a graph of genus n and maximum genus larger than mn.     

Numerical semigroups of genus eight and double coverings of curves of genus three

The authors determine all possible numerical semigroups at ramification points of double coverings of curves when the covered curve is of genus three and the covering curve is of genus eight. Moreover, it is shown that all of such numerical semigroups are actually of double covering type.     

Quaternionic compositum genus

A quaternionic field over the rationals contains three quadratic subfields with a compositum genus relation of the type described in the author's paper in Volume 9 of this journal, involving the representation of a prime as norm in these subfields. These representations had previously been only partially exlored by the transfer of class structure from the rational to the quadratic fields. Here a full exposition is given by constructing the Artin characters when the subfields are $\text{Q}$(2^1/2), $\text{Q}$(q^1/2), and $\text{Q}$(2q)^1/2 (q prime). A special role belongs to q = A² + 32b².     

Knots of genus one or on the number of alternating knots of given genus

We prove that any non-hyperbolic genus one knot except the trefoil does not have a minimal canonical Seifert surface and that there are only polynomially many in the crossing number positive knots of given genus or given unknotting number.

     

Construction of hermitian modular forms of genus 2 from cusp forms of genus 1

One constructs an integral operator, mapping the cusp modular forms of one variable into modular forms relative to Hermitian groups of genus 2 over an imaginary quadratic field. One computes explicitly the Fourier coefficients of the obtained forms.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 51–67, 1985.     

A relative maximum genus graph embedding and its local maximum genus

A relative embedding of a connected graph is an embedding of the graph in some surface with respect to some closed walks, each of which bounds a face of the embedding. The relative maximum genus of a connected graph is the maximum of integerk with the property that the graph has a relative embedding in the orientable surface withk handles. A polynomial algorithm is provided for constructing relative maximum genus embedding of a graph if the relative tree of the graph is planar. Under this condition, just like maximum genus embedding, a graph does not have any locally strict maximum genus.     

Exponentially many maximum genus embeddings and genus embeddings for complete graphs

There are many results on the maximum genus, among which most are written for the existence of values of such embeddings, and few attention has been paid to the estimation of such embeddings and their applications. In this paper we study the number of maximum genus embeddings for a graph and find an exponential lower bound for such numbers. Our results show that in general case, a simple connected graph has exponentially many distinct maximum genus embeddings. In particular, a connected cubic graph G of order n always has at least distinct maximum genus embeddings, where α and m denote, respectively, the number of inner vertices and odd components of an optimal tree T. What surprise us most is that such two extremal embeddings (i.e., the maximum genus embeddings and the genus embeddings) are sometimes closely related with each other. In fact, as applications, we show that for a sufficient large natural number n, there are at least many genus embeddings for complete graph K _n with n ≡ 4, 7, 10 (mod12), where C is a constance depending on the value of n of residue 12. These results improve the bounds obtained by Korzhik and Voss and the methods used here are much simpler and straight. This work was supported by the National Natural Science Foundation of China (Grant No. 10671073), Science and Technology commission of Shanghai Municipality (Grant No. 07XD14011) and Shanghai Leading Academic Discipline Project (Grant No. B407)     

Linear systems of arbitrary genus

Kalman's theory is extended to a complete, smooth, and irreducible algebraic curve of arbitrary genus.     

The nonadditivity of the genus

A class of cubic graphs is introduced for which the genus is a nonadditive function of the genus of subgraphs. This provides a small (28 node) counterexample to Duke’s conjecture concerning the relation of the Betti number to the genus of a graph.     

Strong embeddings of minimum genus

A “folklore conjecture, probably due to Tutte” (as described in [P.D. Seymour, Sums of circuits, in: Graph Theory and Related Topics (Proc. Conf., Univ. Waterloo, 1977), Academic Press, 1979, pp. 341-355]) 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 1970s. 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.     

The genus of space curves

LetC be a curve contained in ℙ _k ³ (k of any characteristic), which is locally Cohen-Macaulay, not contained in a plane and of degreed. We prove thatp _a (C)≤≤((d−2)(d−3))/2. Moreover we show existence of curves with anyd, p _a satisfying this inequality and we characterize those curves for which equality holds.

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Sunto Si dimostra che seC⊃ℙ _k ³ (k di caratteristica qualunque) é una curva, non piana, localmente Cohen-Macaulay, di gradod, allorap _a (C)≤((d−2)(d−3))/2. Si mostra che questa limitazione è ottimale, e si classificano le curve di genere aritmetico massimale.

     

Archimedean solids of genus two

The problem of classifying orientable vertex-transitive maps on a surface with genus two is considered. We construct and classify all simple orientable vertex-transitive maps, with face width at least 3 which can be viewed as generalisations of classical Archimedean solids. The proof is computer-aided. The developed method applies to higher genera as well.     

Modular curves of genus 2

We prove that there are exactly genus two curves defined over such that there exists a nonconstant morphism defined over and the jacobian of is -isogenous to the abelian variety attached by Shimura to a newform . We determine the corresponding newforms and present equations for all these curves.

     

The genus of a module

Let R be a Dedekind domain satisfying the Jordan-Zassenhaus theorem (e.g., the ring of integers in a number field) and Λ a module finite R-algebra. We extend classical results of Jacobinski, Roiter, and Drozd on orders and lattices. In particular, it is shown that the genus of a finitely generated Λ-module M is finite. Moreover, given M, there exist a positive integer t and a finite extension S of R such that a Λ-module N is the genus of M if and only if M^(t) ? N^(t) if and only if M ? S ? N ? S.     

Holomorphic triples of genus 0

Here we study the relationship between the stability of coherent systems and the stability of holomorphic triples over a curve of arbitrary genus. Moreover we apply these results to study some properties and give some examples of holomorphic triples on the projective line.      

Maximum genus of regular graphs

This paper provides tight lower bounds on the maximum genus of a regular graph in terms of its cycle rank. The main tool is a relatively simple theorem that relates lower bounds with the existence (or non-existence) of induced subgraphs with odd cycle rank that are separated from the rest of the graph by cuts of size at most three. Lower bounds on the maximum genus are obtained by bounding from below the size of these odd subgraphs. As a special case, upper-embeddability of a class of graphs is caused by an absence of such subgraphs. A well-known theorem stating that every 4-edge-connected graph is upper-embeddable is a straightforward corollary of the employed method.