# Recent Activity

## Are vertex minor closed classes chi-bounded? ★★

Author(s): Geelen

Question   Is every proper vertex-minor closed class of graphs chi-bounded?

Keywords: chi-bounded; circle graph; coloring; vertex minor

## Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family of graphs is -bounded if there exists a function so that every satisfies .

Conjecture   For every fixed tree , the family of graphs with no induced subgraph isomorphic to is -bounded.

Keywords: chi-bounded; coloring; excluded subgraph; tree

## 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 ★★

Author(s): Matheson; Tarjan

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 ★★

Author(s): Kostochka; Reed

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

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

## Erdös-Szekeres conjecture ★★★

Author(s): Erdos; Szekeres

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

## 4-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow.

Keywords: minor; nowhere-zero flow; Petersen graph

## 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

## 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.

Keywords: coloring; surface

## 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