# Recent Activity

## Kneser–Poulsen conjecture ★★★

Author(s): Kneser; Poulsen

Conjecture   If a finite set of unit balls in is rearranged so that the distance between each pair of centers does not decrease, then the volume of the union of the balls does not decrease.

Keywords: pushing disks

## Wide partition conjecture ★★

Author(s): Chow; Taylor

Conjecture   An integer partition is wide if and only if it is Latin.

Keywords:

## 3-accessibility of Fibonacci numbers ★★

Author(s): Landman; Robertson

Question   Is the set of Fibonacci numbers 3-accessible?

## Simplexity of the n-cube ★★★

Author(s):

Question   What is the minimum cardinality of a decomposition of the -cube into -simplices?

Keywords: cube; decomposition; simplex

## Crossing sequences ★★

Author(s): Archdeacon; Bonnington; Siran

Conjecture   Let be a sequence of nonnegative integers which strictly decreases until .

Then there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus with crossings, but not with less crossings.

Keywords: crossing number; crossing sequence

## The Crossing Number of the Complete Graph ★★★

Author(s):

The crossing number of is the minimum number of crossings in all drawings of in the plane.

Conjecture

Keywords: complete graph; crossing number

## The Crossing Number of the Hypercube ★★

Author(s): Erdos; Guy

The crossing number of is the minimum number of crossings in all drawings of in the plane.

The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.

Conjecture

Keywords: crossing number; hypercube

## Monochromatic reachability or rainbow triangles ★★★

Author(s): Sands; Sauer; Woodrow

In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.

Problem   Let be a tournament with edges colored from a set of three colors. Is it true that must have either a rainbow directed cycle of length three or a vertex so that every other vertex can be reached from by a monochromatic (directed) path?

Keywords: digraph; edge-coloring; tournament

## Rank vs. Genus ★★★

Author(s): Johnson

Question   Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?

Keywords:

## The Hodge Conjecture ★★★★

Author(s): Hodge

Conjecture   Let be a complex projective variety. Then every Hodge class is a rational linear combination of the cohomology classes of complex subvarieties of .

Keywords: Hodge Theory; Millenium Problems

## 2-accessibility of primes ★★

Author(s): Landman; Robertson

Question   Is the set of prime numbers 2-accessible?

Keywords: monochromatic diffsequences; primes

## Non-edges vs. feedback edge sets in digraphs ★★★

Author(s): Chudnovsky; Seymour; Sullivan

For any simple digraph , we let be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and be the size of the smallest feedback edge set.

Conjecture  If is a simple digraph without directed cycles of length , then .

Keywords: acyclic; digraph; feedback edge set; triangle free

## Tarski's exponential function problem ★★

Author(s): Tarski

Conjecture   Is the theory of the real numbers with the exponential function decidable?

Keywords: Decidability

## Counting 3-colorings of the hex lattice ★★

Author(s): Thomassen

Problem   Find .

Keywords: coloring; Lieb's Ice Constant; tiling; torus

## Dense rational distance sets in the plane ★★★

Author(s): Ulam

Problem   Does there exist a dense set so that all pairwise distances between points in are rational?

Keywords: integral distance; rational distance

## Negative association in uniform forests ★★

Author(s): Pemantle

Conjecture   Let be a finite graph, let , and let be the edge set of a forest chosen uniformly at random from all forests of . Then

Keywords: forest; negative association

## Wall-Sun-Sun primes and Fibonacci divisibility ★★

Author(s):

Conjecture   For any prime , there exists a Fibonacci number divisible by exactly once.

Equivalently:

Conjecture   For any prime , does not divide where is the Legendre symbol.

Keywords: Fibonacci; prime

## Total Colouring Conjecture ★★★

Conjecture   A total coloring of a graph is an assignment of colors to the vertices and the edges of such that every pair of adjacent vertices, every pair of adjacent edges and every vertex and incident edge pair, receive different colors. The total chromatic number of a graph , , equals the minimum number of colors needed in a total coloring of . It is an old conjecture of Behzad that for every graph , the total chromatic number equals the maximum degree of a vertex in , plus one or two. In other words,

Keywords: Total coloring

## Edge Reconstruction Conjecture ★★★

Author(s): Harary

Conjecture

Every simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs

Keywords: reconstruction

## Nearly spanning regular subgraphs ★★★

Author(s): Alon; Mubayi

Conjecture   For every and every positive integer , there exists so that every simple -regular graph with has a -regular subgraph with .

Keywords: regular; subgraph