Analysis

Something like Picard for 1-forms ★★

Author(s): Elsner

Conjecture   Let $ D $ be the open unit disk in the complex plane and let $ U_1,\dots,U_n $ be open sets such that $ \bigcup_{j=1}^nU_j=D\setminus\{0\} $. Suppose there are injective holomorphic functions $ f_j : U_j \to \mathbb{C}, $ $ j=1,\ldots,n, $ such that for the differentials we have $ {\rm d}f_j={\rm d}f_k $ on any intersection $ U_j\cap U_k $. Then those differentials glue together to a meromorphic 1-form on $ D $.

Keywords: Essential singularity; Holomorphic functions; Picard's theorem; Residue of 1-form; Riemann surfaces

Posted by MathOMan
updated February 26th, 2010
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Graph Theory

Odd cycles and low oddness ★★

Author(s):

Conjecture   If in a bridgeless cubic graph $ G $ the cycles of any $ 2 $-factor are odd, then $ \omega(G)\leq 2 $, where $ \omega(G) $ denotes the oddness of the graph $ G $, that is, the minimum number of odd cycles in a $ 2 $-factor of $ G $.

Keywords:

Posted by Gagik
updated December 13th, 2011
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Combinatorics

Beneš Conjecture ★★★

Author(s): Beneš

Let $ E $ be a non-empty finite set. Given a partition $ \bf h $ of $ E $, the stabilizer of $ \bf h $, denoted $ S(\bf h) $, is the group formed by all permutations of $ E $ preserving each block of $ \mathbf h $.

Problem  ($ \star $)   Find a sufficient condition for a sequence of partitions $ {\bf h}_1, \dots, {\bf h}_\ell $ of $ E $ to be complete, i.e. such that the product of their stabilizers $ S({\bf h}_1) S({\bf h}_2) \dots S({\bf h}_\ell) $ is equal to the whole symmetric group $ \frak S(E) $ on $ E $. In particular, what about completeness of the sequence $ \bf h,\delta(\bf h),\dots,\delta^{\ell-1}(\bf h) $, given a partition $ \bf h $ of $ E $ and a permutation $ \delta $ of $ E $?
Conjecture  (Beneš)   Let $ \bf u $ be a uniform partition of $ E $ and $ \varphi $ be a permutation of $ E $ such that $ \bf u\wedge\varphi(\bf u)=\bf 0 $. Suppose that the set $ \big(\varphi S({\bf u})\big)^{n} $ is transitive, for some integer $ n\ge2 $. Then $$ \frak S(E) = \big(\varphi S({\bf u})\big)^{2n-1}. $$

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Posted by Vadim Lioubimov
updated January 3rd, 2010
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