# Random

## Seymour's self-minor conjecture ★★★

Author(s): Seymour

**Conjecture**Every infinite graph is a proper minor of itself.

Keywords: infinite graph; minor

## Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

**Problem**Determine the smallest percolating set for the -neighbour bootstrap process in the hypercube.

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

## Antidirected trees in digraphs ★★

Author(s): Addario-Berry; Havet; Linhares Sales; Reed; Thomassé

An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0.

**Conjecture**Let be a digraph. If , then contains every antidirected tree of order .

Keywords:

## 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

## Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

**Conjecture**is a prime iff

Keywords: primality

## Infinite uniquely hamiltonian graphs ★★

Author(s): Mohar

**Problem**Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree ?

Keywords: hamiltonian; infinite graph; uniquely hamiltonian

## Aharoni-Berger conjecture ★★★

**Conjecture**If are matroids on and for every partition of , then there exists with which is independent in every .

Keywords: independent set; matroid; partition

## Is Skewes' number e^e^e^79 an integer? ★★

Author(s):

**Conjecture**

Skewes' number is not an integer.

Keywords:

## Edge-Colouring Geometric Complete Graphs ★★

Author(s): Hurtado

**Question**What is the minimum number of colours such that every complete geometric graph on vertices has an edge colouring such that:

- \item[Variant A] crossing edges get distinct colours, \item[Variant B] disjoint edges get distinct colours, \item[Variant C] non-disjoint edges get distinct colours, \item[Variant D] non-crossing edges get distinct colours.

Keywords: geometric complete graph, colouring

## Are there an infinite number of lucky primes? ★

Author(s): Lazarus: Gardiner: Metropolis; Ulam

**Conjecture**If every second positive integer except 2 is remaining, then every third remaining integer except 3, then every fourth remaining integer etc. , an infinite number of the remaining integers are prime.

## Every prism over a 3-connected planar graph is hamiltonian. ★★

Author(s): Kaiser; Král; Rosenfeld; Ryjácek; Voss

**Conjecture**If is a -connected planar graph, then has a Hamilton cycle.

Keywords:

## Subdivision of a transitive tournament in digraphs with large outdegree. ★★

Author(s): Mader

**Conjecture**For all there is an integer such that every digraph of minimum outdegree at least contains a subdivision of a transitive tournament of order .

Keywords:

## Every metamonovalued reloid is monovalued ★★

Author(s): Porton

**Conjecture**Every metamonovalued reloid is monovalued.

Keywords:

## The Bermond-Thomassen Conjecture ★★

**Conjecture**For every positive integer , every digraph with minimum out-degree at least contains disjoint cycles.

Keywords: cycles

## End-Devouring Rays ★

Author(s): Georgakopoulos

**Problem**Let be a graph, a countable end of , and an infinite set of pairwise disjoint -rays in . Prove that there is a set of pairwise disjoint -rays that devours such that the set of starting vertices of rays in equals the set of starting vertices of rays in .

## Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and .

A -regular graph is a -graph if for every with odd.

**Conjecture**Let be an integer. If is a -graph, then .

Keywords: flow conjectures; nowhere-zero flows

## Smooth 4-dimensional Schoenflies problem ★★★★

Author(s): Alexander

**Problem**Let be a -dimensional smooth submanifold of , diffeomorphic to . By the Jordan-Brouwer separation theorem, separates into the union of two compact connected -manifolds which share as a common boundary. The Schoenflies problem asks, are these -manifolds diffeomorphic to ? ie: is unknotted?

Keywords: 4-dimensional; Schoenflies; sphere

## ¿Are critical k-forests tight? ★★

Author(s): Strausz

**Conjecture**

Let be a -uniform hypergraph. If is a critical -forest, then it is a -tree.

Keywords: heterochromatic number

## Separators in string graphs ★★

**Conjecture**Every string graph with edges has a separator of size .

Keywords: separator; string graphs

## Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let be a proximity.

A set is connected regarding iff .

**Conjecture**The following statements are equivalent for every endofuncoid and a set :

- \item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .

Keywords: connected; connectedness; proximity space

## Unit vector flows ★★

Author(s): Jain

**Conjecture**For every graph without a bridge, there is a flow .

**Conjecture**There exists a map so that antipodal points of receive opposite values, and so that any three points which are equidistant on a great circle have values which sum to zero.

Keywords: nowhere-zero flow

## Outer reloid of restricted funcoid ★★

Author(s): Porton

**Question**for every filter objects and and a funcoid ?

Keywords: direct product of filters; outer reloid

## Barnette's Conjecture ★★★

Author(s): Barnette

**Conjecture**Every 3-connected cubic planar bipartite graph is Hamiltonian.

Keywords: bipartite; cubic; hamiltonian

## Friendly partitions ★★

Author(s): DeVos

A *friendly* partition of a graph is a partition of the vertices into two sets so that every vertex has at least as many neighbours in its own class as in the other.

**Problem**Is it true that for every , all but finitely many -regular graphs have friendly partitions?

## Decomposing eulerian graphs ★★★

Author(s):

**Conjecture**If is a 6-edge-connected Eulerian graph and is a 2-transition system for , then has a compaible decomposition.

## Coloring and immersion ★★★

Author(s): Abu-Khzam; Langston

**Conjecture**For every positive integer , every (loopless) graph with immerses .

Keywords: coloring; complete graph; immersion

## PTAS for feedback arc set in tournaments ★★

**Question**Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?

Keywords: feedback arc set; PTAS; tournament

## 3-accessibility of Fibonacci numbers ★★

**Question**Is the set of Fibonacci numbers 3-accessible?

Keywords: Fibonacci numbers; monochromatic diffsequences

## 2-colouring a graph without a monochromatic maximum clique ★★

**Conjecture**If is a non-empty graph containing no induced odd cycle of length at least , then there is a -vertex colouring of in which no maximum clique is monochromatic.

Keywords: maximum clique; Partitioning

## Realisation problem for the space of knots in the 3-sphere ★★

Author(s): Budney

**Problem**Given a link in , let the symmetry group of be denoted ie: isotopy classes of diffeomorphisms of which preserve , where the isotopies are also required to preserve .

Now let be a hyperbolic link. Assume has the further `Brunnian' property that there exists a component of such that is the unlink. Let be the subgroup of consisting of diffeomorphisms of which preserve together with its orientation, and which preserve the orientation of .

There is a representation given by restricting the diffeomorphism to the . It's known that is always a cyclic group. And is a signed symmetric group -- the wreath product of a symmetric group with .

Problem: What representations can be obtained?

Keywords: knot space; symmetry

## Minimum number of arc-disjoint transitive subtournaments of order 3 in a tournament ★★

Author(s): Yuster

**Conjecture**If is a tournament of order , then it contains arc-disjoint transitive subtournaments of order 3.

Keywords:

## Partial List Coloring ★★★

Author(s): Iradmusa

Let be a simple graph, and for every list assignment let be the maximum number of vertices of which are colorable with respect to . Define , where the minimum is taken over all list assignments with for all .

**Conjecture**[2] Let be a graph with list chromatic number and . Then

Keywords: list assignment; list coloring

## Every 4-connected toroidal graph has a Hamilton cycle ★★

Author(s): Grunbaum; Nash-Williams

**Conjecture**Every 4-connected toroidal graph has a Hamilton cycle.

Keywords:

## Concavity of van der Waerden numbers ★★

Author(s): Landman

For and positive integers, the (mixed) van der Waerden number is the least positive integer such that every (red-blue)-coloring of admits either a -term red arithmetic progression or an -term blue arithmetic progression.

**Conjecture**For all and with , .

Keywords: arithmetic progression; van der Waerden

## 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

## Monochromatic empty triangles ★★★

Author(s):

If is a finite set of points which is 2-colored, an *empty triangle* is a set with so that the convex hull of is disjoint from . We say that is *monochromatic* if all points in are the same color.

**Conjecture**There exists a fixed constant with the following property. If is a set of points in general position which is 2-colored, then it has monochromatic empty triangles.

Keywords: empty triangle; general position; ramsey theory

## Graphs of exact colorings ★★

Author(s):

Conjecture For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .

Keywords:

## F_d versus F_{d+1} ★★★

Author(s): Krajicek

**Problem**Find a constant such that for any there is a sequence of tautologies of depth that have polynomial (or quasi-polynomial) size proofs in depth Frege system but requires exponential size proofs.

Keywords: Frege system; short proof

## Universal point sets for planar graphs ★★★

Author(s): Mohar

We say that a set is -*universal* if every vertex planar graph can be drawn in the plane so that each vertex maps to a distinct point in , and all edges are (non-intersecting) straight line segments.

**Question**Does there exist an -universal set of size ?

Keywords: geometric graph; planar graph; universal set

## Turán's problem for hypergraphs ★★

Author(s): Turan

**Conjecture**Every simple -uniform hypergraph on vertices which contains no complete -uniform hypergraph on four vertices has at most hyperedges.

**Conjecture**Every simple -uniform hypergraph on vertices which contains no complete -uniform hypergraph on five vertices has at most hyperedges.

Keywords:

## Goldberg's conjecture ★★★

Author(s): Goldberg

The *overfull parameter* is defined as follows:

**Conjecture**Every graph satisfies .

Keywords: edge-coloring; multigraph

## $C^r$ Stability Conjecture ★★★★

**Conjecture**Any structurally stable diffeomorphism is hyperbolic.

Keywords: diffeomorphisms,; dynamical systems

## Roller Coaster permutations ★★★

Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .

A permutation is called a *Roller Coaster permutation* if . Let be the set of all Roller Coaster permutations in .

**Conjecture**For ,

- \item If , then . \item If , then with .

**Conjecture (Odd Sum conjecture)**Given ,

- \item If , then is odd for . \item If , then for all .

Keywords:

## Kriesell's Conjecture ★★

Author(s): Kriesell

**Conjecture**Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .

Keywords: Disjoint paths; edge-connectivity; spanning trees

## Weak pentagon problem ★★

Author(s): Samal

**Conjecture**If is a cubic graph not containing a triangle, then it is possible to color the edges of by five colors, so that the complement of every color class is a bipartite graph.

Keywords: Clebsch graph; cut-continuous mapping; edge-coloring; homomorphism; pentagon

## Transversal achievement game on a square grid ★★

Author(s): Erickson

**Problem**Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an grid. The first player (if any) to occupy a set of cells having no two cells in the same row or column is the winner. What is the outcome of the game given optimal play?

Keywords: game

## Ádám's Conjecture ★★★

Author(s): Ádám

**Conjecture**Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles.

Keywords:

## The stubborn list partition problem ★★

Author(s): Cameron; Eschen; Hoang; Sritharan

**Problem**Does there exist a polynomial time algorithm which takes as input a graph and for every vertex a subset of , and decides if there exists a partition of into so that only if and so that are independent, is a clique, and there are no edges between and ?

Keywords: list partition; polynomial algorithm

## Continous analogue of Hirsch conjecture ★★

Author(s): Deza; Terlaky; Zinchenko

**Conjecture**The order of the largest total curvature of the primal central path over all polytopes defined by inequalities in dimension is .