# Random

## Number of Cliques in Minor-Closed Classes ★★

Author(s): Wood

**Question**Is there a constant such that every -vertex -minor-free graph has at most cliques?

## Partition of Complete Geometric Graph into Plane Trees ★★

Author(s):

**Conjecture**Every complete geometric graph with an even number of vertices has a partition of its edge set into plane (i.e. non-crossing) spanning trees.

Keywords: complete geometric graph, edge colouring

## Hoàng-Reed Conjecture ★★★

**Conjecture**Every digraph in which each vertex has outdegree at least contains directed cycles such that meets in at most one vertex, .

Keywords:

## Convex Equipartitions with Extreme Perimeter ★★

Author(s): Nandakumar

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Keywords: convex equipartition

## 3-accessibility of Fibonacci numbers ★★

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

Keywords: Fibonacci numbers; monochromatic diffsequences

## Every metamonovalued funcoid is monovalued ★★

Author(s): Porton

**Conjecture**Every metamonovalued funcoid is monovalued.

The reverse is almost trivial: Every monovalued funcoid is metamonovalued.

Keywords: monovalued

## 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

## Are different notions of the crossing number the same? ★★★

**Problem**Does the following equality hold for every graph ?

The crossing number of a graph is the minimum number of edge crossings in any drawing of in the plane. In the *pairwise crossing number* , we minimize the number of pairs of edges that cross.

Keywords: crossing number; pair-crossing number

## 4-regular 4-chromatic graphs of high girth ★★

Author(s): Grunbaum

**Problem**Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth?

## The 4x5 chessboard complex is the complement of a link, which link? ★★

Author(s): David Eppstein

**Problem**Ian Agol and Matthias Goerner observed that the 4x5 chessboard complex is the complement of many distinct links in the 3-sphere. Their observation is non-constructive, as it uses the resolution of the Poincare Conjecture. Find specific links that have the 4x5 chessboard complex as their complement.

Keywords: knot theory, links, chessboard complex

## Earth-Moon Problem ★★

Author(s): Ringel

**Problem**What is the maximum number of colours needed to colour countries such that no two countries sharing a common border have the same colour in the case where each country consists of one region on earth and one region on the moon ?

Keywords:

## Nonseparating planar continuum ★★

Author(s):

**Conjecture**Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point.

Keywords: fixed point

## Edge-disjoint Hamilton cycles in highly strongly connected tournaments. ★★

Author(s): Thomassen

**Conjecture**For every , there is an integer so that every strongly -connected tournament has edge-disjoint Hamilton cycles.

Keywords:

## Real roots of the flow polynomial ★★

Author(s): Welsh

**Conjecture**All real roots of nonzero flow polynomials are at most 4.

Keywords: flow polynomial; nowhere-zero flow

## Are there infinite number of Mersenne Primes? ★★★★

Author(s):

**Conjecture**A Mersenne prime is a Mersenne number that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime

## Choice Number of k-Chromatic Graphs of Bounded Order ★★

Author(s): Noel

**Conjecture**If is a -chromatic graph on at most vertices, then .

Keywords: choosability; complete multipartite graph; list coloring

## Forcing a 2-regular minor ★★

**Conjecture**Every graph with average degree at least contains every 2-regular graph on vertices as a minor.

Keywords: minors

## Special Primes ★

Author(s): George BALAN

**Conjecture**Let be a prime natural number. Find all primes , such that .

Keywords:

## Long rainbow arithmetic progressions ★★

Author(s): Fox; Jungic; Mahdian; Nesetril; Radoicic

For let denote the minimal number such that there is a rainbow in every equinumerous -coloring of for every

**Conjecture**For all , .

Keywords: arithmetic progression; rainbow

## List chromatic number and maximum degree of bipartite graphs ★★

Author(s): Alon

**Conjecture**There is a constant such that the list chromatic number of any bipartite graph of maximum degree is at most .

Keywords:

## Decomposition of completions of reloids ★★

Author(s): Porton

**Conjecture**For composable reloids and it holds

- \item if is a co-complete reloid; \item if is a complete reloid; \item ; \item ; \item .

Keywords: co-completion; completion; reloid

## Hamilton cycle in small d-diregular graphs ★★

Author(s): Jackson

An directed graph is -diregular if every vertex has indegree and outdegree at least .

**Conjecture**For , every -diregular oriented graph on at most vertices has a Hamilton cycle.

Keywords:

## Highly connected graphs with no K_n minor ★★★

Author(s): Thomas

**Problem**Is it true for all , that every sufficiently large -connected graph without a minor has a set of vertices whose deletion results in a planar graph?

Keywords: connectivity; minor

## Lindelöf hypothesis ★★

Author(s): Lindelöf

**Conjecture**For any

Since can be replaced by a smaller value, we can also write the conjecture as, for any positive ,

Keywords: Riemann Hypothesis; zeta

## Weak saturation of the cube in the clique ★

**Problem**

Determine .

Keywords: bootstrap percolation; hypercube; Weak saturation

## 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

What about

?

Keywords: surreal numbers

## Ramsey properties of Cayley graphs ★★★

Author(s): Alon

**Conjecture**There exists a fixed constant so that every abelian group has a subset with so that the Cayley graph has no clique or independent set of size .

Keywords: Cayley graph; Ramsey number

## The Bermond-Thomassen Conjecture ★★

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

Keywords: cycles

## Sequence defined on multisets ★★

Author(s): Erickson

**Conjecture**Define a array of positive integers where the first row consists of some distinct positive integers arranged in increasing order, and the second row consists of any positive integers in any order. Create a new array where the first row consists of all the integers that occur in the first array, arranged in increasing order, and the second row consists of their multiplicities. Repeat the process. For example, starting with the array , the sequence is: -> -> -> -> -> -> -> -> -> -> -> , and we now have a fixed point (loop of one array).

The process always results in a loop of 1, 2, or 3 arrays.

## Odd incongruent covering systems ★★★

**Conjecture**There is no covering system whose moduli are odd, distinct, and greater than 1.

Keywords: covering system

## What is the smallest number of disjoint spanning trees made a graph Hamiltonian ★★

Author(s): Goldengorin

We are given a complete simple undirected weighted graph and its first arbitrary shortest spanning tree . We define the next graph and find on the second arbitrary shortest spanning tree . We continue similarly by finding on , etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let be the graph obtained as union of all disjoint trees.

**Question 1**. What is the smallest number of disjoint spanning trees creates a graph containing a Hamiltonian path.

**Question 2**. What is the smallest number of disjoint spanning trees creates a graph containing a shortest Hamiltonian path?

**Questions 3 and 4**. Replace in questions 1 and 2 a shortest spanning tree by a 1-tree. What is the smallest number of disjoint 1-trees creates a Hamiltonian graph? What is the smallest number of disjoint 1-trees creates a graph containing a shortest Hamiltonian cycle?

Keywords: 1-trees; cycle; Hamitonian path; spanning trees

## Are all Mersenne Numbers with prime exponent square-free? ★★★

Author(s):

**Conjecture**Are all Mersenne Numbers with prime exponent Square free?

Keywords: Mersenne number

## Switching reconstruction of digraphs ★★

**Question**Are there any switching-nonreconstructible digraphs on twelve or more vertices?

Keywords:

## Vertex Coloring of graph fractional powers ★★★

Author(s): Iradmusa

**Conjecture**Let be a graph and be a positive integer. The

*power of*, denoted by , is defined on the vertex set , by connecting any two distinct vertices and with distance at most . In other words, . Also

*subdivision of*, denoted by , is constructed by replacing each edge of with a path of length . Note that for , we have .

Now we can define the fractional power of a graph as follows:

Let be a graph and . The graph is defined by the power of the subdivision of . In other words .

*Conjecture.*Let be a connected graph with and be a positive integer greater than 1. Then for any positive integer , we have .

In [1], it was shown that this conjecture is true in some special cases.

Keywords: chromatic number, fractional power of graph, clique number

## 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

## List Total Colouring Conjecture ★★

Author(s): Borodin; Kostochka; Woodall

**Conjecture**If is the total graph of a multigraph, then .

Keywords: list coloring; Total coloring; total graphs

## Distribution and upper bound of mimic numbers ★★

Author(s): Bhattacharyya

**Problem**

Let the notation denote '' divides ''. The mimic function in number theory is defined as follows [1].

**Definition**For any positive integer divisible by , the mimic function, , is given by,

By using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].

**Definition**The number is defined to be the mimic number of any positive integer , with respect to , for the minimum value of which .

Given these two definitions and a positive integer , find the distribution of mimic numbers of those numbers divisible by .

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer .

Keywords: Divisibility; mimic function; mimic number

## Cyclic spanning subdigraph with small cyclomatic number ★★

Author(s): Bondy

**Conjecture**Let be a digraph all of whose strong components are nontrivial. Then contains a cyclic spanning subdigraph with cyclomatic number at most .

Keywords:

## Cores of Cayley graphs ★★★★★

Author(s): Samal

**Conjecture**Let be an abelian group. Is the core of a Cayley graph (on some power of ) a Cayley graph (on some power of )?

Keywords: Cayley graph; core

## 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

## Approximation ratio for k-outerplanar graphs ★★

Author(s): Bentz

**Conjecture**Is the approximation ratio for the

*Maximum Edge Disjoint Paths*(MaxEDP) or the

*Maximum Integer Multiflow*problem (MaxIMF) bounded by a constant in -outerplanar graphs or tree-width graphs?

Keywords: approximation algorithms; planar graph; polynomial algorithm

## Universal highly arc transitive digraphs ★★★

Author(s): Cameron; Praeger; Wormald

An *alternating* walk in a digraph is a walk so that the vertex is either the head of both and or the tail of both and for every . A digraph is *universal* if for every pair of edges , there is an alternating walk containing both and

**Question**Does there exist a locally finite highly arc transitive digraph which is universal?

Keywords: arc transitive; digraph

## Strong matchings and covers ★★★

Author(s): Aharoni

Let be a hypergraph. A *strongly maximal* matching is a matching so that for every matching . A *strongly minimal* cover is a (vertex) cover so that for every cover .

**Conjecture**If is a (possibly infinite) hypergraph in which all edges have size for some integer , then has a strongly maximal matching and a strongly minimal cover.

Keywords: cover; infinite graph; matching

## Birch & Swinnerton-Dyer conjecture ★★★★

Author(s):

**Conjecture**Let be an elliptic curve over a number field . Then the order of the zeros of its -function, , at is the Mordell-Weil rank of .

Keywords:

## Rendezvous on a line ★★★

Author(s): Alpern

**Problem**Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum speed of 1. What is the infimum expected meeting time (first time when the players occupy the same point) which can be achieved assuming the two players must adopt the same strategy?

Keywords: game theory; optimization; rendezvous

## Fat 4-polytopes ★★★

Author(s): Eppstein; Kuperberg; Ziegler

The *fatness* of a 4-polytope is defined to be where is the number of faces of of dimension .

**Question**Does there exist a fixed constant so that every convex 4-polytope has fatness at most ?

## What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let be an indexed family of sets.

*Products* are for .

*Hyperfuncoids* are filters on the lattice of all finite unions of products.

**Problem**Is a bijection from hyperfuncoids to:

- \item prestaroids on ; \item staroids on ; \item completary staroids on ?

If yes, is defining the inverse bijection? If not, characterize the image of the function defined on .

Consider also the variant of this problem with the set replaced with the set of complements of elements of the set .

Keywords: hyperfuncoids; multidimensional

## The Bollobás-Eldridge-Catlin Conjecture on graph packing ★★★

Author(s):

**Conjecture (BEC-conjecture)**If and are -vertex graphs and , then and pack.

Keywords: graph packing

## 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

## Rota's unimodal conjecture ★★★

Author(s): Rota

Let be a matroid of rank , and for let be the number of closed sets of rank .

**Conjecture**is unimodal.

**Conjecture**is log-concave.

Keywords: flat; log-concave; matroid