# Recent Activity

## Asymptotic Distribution of Form of Polyhedra ★★

Author(s): Rüdinger

**Problem**Consider the set of all topologically inequivalent polyhedra with edges. Define a form parameter for a polyhedron as where is the number of vertices. What is the distribution of for ?

Keywords: polyhedral graphs, distribution

## Domination in plane triangulations ★★

**Conjecture**Every sufficiently large plane triangulation has a dominating set of size .

Keywords: coloring; domination; multigrid; planar graph; triangulation

## Bounding the chromatic number of triangle-free graphs with fixed maximum degree ★★

**Conjecture**A triangle-free graph with maximum degree has chromatic number at most .

Keywords: chromatic number; girth; maximum degree; triangle free

## Erdös-Szekeres conjecture ★★★

**Conjecture**Every set of points in the plane in general position contains a subset of points which form a convex -gon.

Keywords: combinatorial geometry; Convex Polygons; ramsey theory

## Inequality of the means ★★★

Author(s):

**Question**Is is possible to pack rectangular -dimensional boxes each of which has side lengths inside an -dimensional cube with side length ?

Keywords: arithmetic mean; geometric mean; Inequality; packing

## P vs. PSPACE ★★★

Author(s): Folklore

**Problem**Is there a problem that can be computed by a Turing machine in polynomial space and unbounded time but not in polynomial time? More formally, does P = PSPACE?

Keywords: P; PSPACE; separation; unconditional

## Sums of independent random variables with unbounded variance ★★

Author(s): Feige

**Conjecture**If are independent random variables with , then

Keywords: Inequality; Probability Theory; randomness in TCS

## Grunbaum's Conjecture ★★★

Author(s): Grunbaum

**Conjecture**If is a simple loopless triangulation of an orientable surface, then the dual of is 3-edge-colorable.

## Refuting random 3SAT-instances on $O(n)$ clauses (weak form) ★★★

Author(s): Feige

**Conjecture**For every rational and every rational , there is no polynomial-time algorithm for the following problem.

Given is a 3SAT (3CNF) formula on variables, for some , and clauses drawn uniformly at random from the set of formulas on variables. Return with probability at least 0.5 (over the instances) that is *typical* without returning *typical* for *any* instance with at least simultaneously satisfiable clauses.

Keywords: NP; randomness in TCS; satisfiability

## Does the chromatic symmetric function distinguish between trees? ★★

Author(s): Stanley

**Problem**Do there exist non-isomorphic trees which have the same chromatic symmetric function?

Keywords: chromatic polynomial; symmetric function; tree

## Shannon capacity of the seven-cycle ★★★

Author(s):

**Problem**What is the Shannon capacity of ?

Keywords:

## Book Thickness of Subdivisions ★★

Author(s): Blankenship; Oporowski

Let be a finite undirected simple graph.

A *-page book embedding* of consists of a linear order of and a (non-proper) -colouring of such that edges with the same colour do not cross with respect to . That is, if for some edges , then and receive distinct colours.

One can think that the vertices are placed along the spine of a book, and the edges are drawn without crossings on the pages of the book.

The *book thickness* of , denoted by bt is the minimum integer for which there is a -page book embedding of .

Let be the graph obtained by subdividing each edge of exactly once.

**Conjecture**There is a function such that for every graph ,

Keywords: book embedding; book thickness

## Frobenius number of four or more integers ★★

Author(s):

**Problem**Find an explicit formula for Frobenius number of co-prime positive integers for .

Keywords:

## Magic square of squares ★★

Author(s): LaBar

**Question**Does there exist a magic square composed of distinct perfect squares?

Keywords:

## Diophantine quintuple conjecture ★★

Author(s):

**Definition**A set of m positive integers is called a Diophantine -tuple if is a perfect square for all .

**Conjecture (1)**Diophantine quintuple does not exist.

It would follow from the following stronger conjecture [Da]:

**Conjecture (2)**If is a Diophantine quadruple and , then

Keywords:

## Inverse Galois Problem ★★★★

Author(s): Hilbert

**Conjecture**Every finite group is the Galois group of some finite algebraic extension of .

Keywords:

## Seymour's r-graph conjecture ★★★

Author(s): Seymour

An -*graph* is an -regular graph with the property that for every with odd size.

**Conjecture**for every -graph .

Keywords: edge-coloring; r-graph

## Edge list coloring conjecture ★★★

Author(s):

**Conjecture**Let be a loopless multigraph. Then the edge chromatic number of equals the list edge chromatic number of .

Keywords:

## Middle levels problem ★★

Author(s): Erdos

**Conjecture**Let be the bipartite graph whose vertices are the -subsets and the -subsets of a -element set, and with inclusion as the adjacency relationship. Then is Hamiltonian.

Keywords: