Recent Activity

Euler-Mascheroni constant ★★★

Author(s):

Question   Is Euler-Mascheroni constant an transcendental number?

Keywords: constant; Euler; irrational; Mascheroni; rational; transcendental

Graham's conjecture on tree reconstruction ★★

Author(s): Graham

Problem   for every graph , we let denote the line graph of . Given that is a tree, can we determine it from the integer sequence ?

Keywords: reconstruction; tree

Vertex Cover Integrality Gap ★★

Author(s): Atserias

Conjecture   For every there is such that, for every large , there are -vertex graphs and such that and .

Keywords: counting quantifiers; FMT12-LesHouches

Big Line or Big Clique in Planar Point Sets ★★

Author(s): Kara; Por; Wood

Let be a set of points in the plane. Two points and in are visible with respect to if the line segment between and contains no other point in .

Conjecture   For all integers there is an integer such that every set of at least points in the plane contains at least collinear points or pairwise visible points.

Mixing Circular Colourings ★

Author(s): Brewster; Noel

Question   Is always rational?

Keywords: discrete homotopy; graph colourings; mixing

Finite entailment of Positive Horn logic ★★

Author(s): Martin

Question   Positive Horn logic (pH) is the fragment of FO involving exactly . Does the fragment have the finite model property?

Keywords: entailment; finite satisfiability; horn logic

The Borodin-Kostochka Conjecture ★★

Author(s): Borodin; Kostochka

Conjecture   Every graph with maximum degree has chromatic number at most .

Keywords:

Chromatic number of random lifts of complete graphs ★★

Author(s): Amit

Question   Is the chromatic number of a random lift of concentrated on a single value?

Keywords: random lifts, coloring

3 is a primitive root modulo primes of the form 16 q^4 + 1, where q>3 is prime ★★

Author(s):

Conjecture   is a primitive root modulo for all primes , where is prime.

Keywords:

Circular choosability of planar graphs ★

Author(s): Mohar

Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is

Problem   What is the best upper bound on circular choosability for planar graphs?

Keywords: choosability; circular colouring; planar graphs

A conjecture about direct product of funcoids ★★

Author(s): Porton

Conjecture   Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

Keywords: category theory; general topology

Criterion for boundedness of power series ★

Author(s): Rüdinger

Question   Give a necessary and sufficient criterion for the sequence so that the power series is bounded for all .

Keywords: boundedness; power series; real analysis

Length of surreal product ★

Author(s): Gonshor

Conjecture   Every surreal number has a unique sign expansion, i.e. function , where is some ordinal. This is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of as .

It is easy to prove that

?

Keywords: surreal numbers

Durer's Conjecture ★★★

Author(s): Durer; Shephard

Conjecture   Every convex polytope has a non-overlapping edge unfolding.

Keywords: folding; polytope

Convex uniform 5-polytopes ★★

Author(s):

Problem   Enumerate all convex uniform 5-polytopes.

Keywords:

MSO alternation hierarchy over pictures ★★

Author(s): Grandjean

Question   Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linearly) related.

Blatter-Specker Theorem for ternary relations ★★

Author(s): Makowsky

Let be a class of finite relational structures. We denote by the number of structures in over the labeled set . For any class definable in monadic second-order logic with unary and binary relation symbols, Specker and Blatter showed that, for every , the function is ultimately periodic modulo .

Question   Does the Blatter-Specker Theorem hold for ternary relations.

Keywords: Blatter-Specker Theorem; FMT00-Luminy

Monadic second-order logic with cardinality predicates ★★

Author(s): Courcelle

The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:

\item \item

where is a fixed recursive set of integers.

Let us fix and a closed formula in this language.

Conjecture   Is it true that the validity of for a graph of tree-width at most can be tested in polynomial time in the size of ?

Order-invariant queries ★★

Author(s): Segoufin

Question
\item Does hold over graphs of bounded tree-width? \item Is included in over graphs? \item Does have a 0-1 law? \item Are properties of Hanf-local? \item Is there a logic (with an effective syntax) that captures ?

Fixed-point logic with counting ★★

Author(s): Blass

Question   Can either of the following be expressed in fixed-point logic plus counting:
\item Given a graph, does it have a perfect matching, i.e., a set of edges such that every vertex is incident to exactly one edge from ? \item Given a square matrix over a finite field (regarded as a structure in the natural way, as described in [BGS02]), what is its determinant?