Counterexamples to a conjecture by Gross,Mansour and Tucker on partial-dual genus polynomials of ribbon graphs

Gross, Mansour and Tucker introduced the partial-dual orientable genus polynomial and the partial-dual Euler genus polynomial. They computed these two partial-dual genus polynomials of four families of ribbon graphs, posed some research problems and made some conjectures. In this paper, we introduce the notion of signed interlace sequences of bouquets and obtain the partial-dual Euler genus polynomials for all ribbon graphs with the number of edges less than 4 and the partial-dual orientable genus polynomials for all orientable ribbon graphs with the number of edges less than 5 in terms of signed interlace sequences. We check all the conjectures and find a counterexample to the Conjecture 3.1 in their paper: There is no orientable ribbon graph having a non-constant partial-dual genus polynomial with only one non-zero coefficient. Motivated by this counterexample, we further find an infinite family of counterexamples to the conjecture.     

On an analogue of a conjecture of Gross

The paper is concerned with a problem in the theory of congruence function fields which is analogous to a conjecture of Gross in Iwasawa Theory. Z_p-extensions K_∞/K₀ of congruence function fields K₀ of characteristic p≠2 involving no new constants are considered such that the set S of ramified primes is finite and these primes are fully ramified. Is the set of S-classes invariant under Gal(K_∞/K₀) finite ? Gross' conjecture asserts that a similar question has an affirmative answer for the class of cyclotomic Z_p- extensions of CM-type if S is the set of p-primes and the classes considered are minus S-classes. Using a formula of Witt for the norm residue symbol in cyclic p-extensions of local fields of characteristic p, a necessary and sufficient condition for the validity of the analogue of Gross' conjecture is given for a class of extensions K_∞/K₀. It is shown by examples that the analogue of Gross' conjecture is not always true.     

Proof of a conjecture of Gross concerning fix-points

Research performed as a Feodor Lynen Research Fellow of the Alexander von Humboldt Foundation at Cornell University     

The Gross conjecture over rational function fields

We study the Gross conjecture for the cyclotomic function field extension k(∧f)/k where k = Fq(t) is the rational function field and f is a monic polynomial in Fq[t]. We prove the conjecture in the Fermat curve case(i.e., when f = t(t - 1)) by a direct calculation. We also prove the case when f is irreducible, which is analogous to the Weil reciprocity law. In the general case, we manage to show the weak version of the Gross conjecture here.     

On a Chang conjecture

We use the core model for the one strong cardinal to show that the Chang Conjecture (ℵ _n+2, ℵ _n+1) ⇒ (ℵ _n+1, ℵ _n ) together with 2^ℵ _{n − 1} = ℵ _n implies, for 1<n<ω, the existence of an inner model with a strong cardinal. An essential step of our proof is an application of the Gitik Game which also admits a presentation.     

On a conjecture of Vorst

Let k be an infinite perfect field of positive characteristic such that strong resolution of singularities holds over k. We prove that a localization of a d-dimensional commutative k-algebra R of finite type is K _d+1-regular if and only if it is regular. This partially affirms a conjecture of Vorst.     

On a conjecture of Lynch

Abstract

We comment on a conjecture of Lynch on annihilators of local cohomology.

Communicated by Lawrence Ein     

On a conjecture of Bredon

Using the localization technique for equivariant cohomology theory we prove a conjecture of G.Bredon (s. [4], p.381) which states that under certain conditions (s. the theorem below) the cohomology with Z_p-coefficients of each component of the fixpoint set of a Z_p-space can be generated as an algebra by (at most) the same number of elements as the cohomology of the space itself.     

On a conjecture of Alon

Text

Let f(n,m) be the cardinality of largest subset of {1,2,…,n} which does not contain a subset whose elements sum to m. In this note, we show that

     

On a conjecture of Zaremba

LetN _C (x) be the number of integersm≤x such that there is an integera with 1≤a<m, (a, m)=1 and all partial quotients in the continued fraction expansion ofa/m are at mostC. We prove for allx≥1 that $$N_c (x) > {1 \mathord{\left/ {\vphantom {1 {\sqrt {2C} x^{{1 \mathord{\left/ {\vphantom {1 {2(1 - 1/C^2 )}}} \right. \kern-\nulldelimiterspace} {2(1 - 1/C^2 )}}} }}} \right. \kern-\nulldelimiterspace} {\sqrt {2C} x^{{1 \mathord{\left/ {\vphantom {1 {2(1 - 1/C^2 )}}} \right. \kern-\nulldelimiterspace} {2(1 - 1/C^2 )}}} }}$$ .     

On a conjecture of Meyniel

A graph G is said to be very strongly perfect if for each induced subgraph H of G, each vertex of H belongs to a stable set that meets all maximal cliques of H. Meyniel proved that a graph is perfect if each of its odd cycles with at least five vertices contains at least two chords. Nowadays, such a graph is called a Meyniel graph. We prove that, as conjectured by Meyniel, a graph is very strongly perfect if and only if it is a Meyniel graph. We also design a polynomial-time algorithm which, given a Meyniel graph G and a vertex x of G, finds a stable set that contains x and meets all maximal cliques of G. We shall convert this algorithm into another polynomial-time algorithm which, given a Meyniel graph G, finds an optimal coloring of G, and a largest clique of G. Finally, we shall establish another property, related to perfection, of Meyniel graphs.     

On a conjecture of ridge

The conjecture of Ridge on the numerical range of a shift of periodic weights is resolved in the affirmative, i.e., if the weights are nonzero, the numerical range of the corresponding shift is an open disc centered at the origin. The radius of the disc can be expressed as the Perron root of a nonnegative irreducible symmetric matrix. Some related results are obtained.

     

On a conjecture of Broughan

In this paper, we confirm a conjecture of Broughan, concerning the closure of the set of Fibonacci numbers in the full topology over .

     

On a conjecture of bondy

Bondy conjectured [1] that: if G is a k-connected graph, where k ≥ 2, such that the degree-sum of any k + 1 independent vertices is at least m, then G contains a cycle of length at least: $\text{Min}\text{(}\text{2m}\text{(k + 1)}\text{, n)}$ (n denotes the order of G). We prove here that this result is true.