# 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?

Keywords: clique; graph; minor

## 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.

## Hoàng-Reed Conjecture ★★★

Author(s): Hoang; Reed

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 ★★

Author(s): Landman; Robertson

Question   Is the set of Fibonacci numbers 3-accessible?

## 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? ★★★

Author(s): Pach; Tóth

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?

Keywords: coloring; 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.

## 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 .

## Forcing a 2-regular minor ★★

Author(s): Reed; Wood

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 ★

Author(s): Morrison; Noel

Problem

Determine .

## 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

?

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 ★★

Author(s): Bermond; Thomassen

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.

Keywords: multiset; sequence

## Odd incongruent covering systems ★★★

Author(s): Erdos; Selfridge

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 ★★

Author(s): Bondy; Mercier

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

Keywords:

## Vertex Coloring of graph fractional powers ★★★

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.

## 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?

## 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 ?

Keywords: f-vector; polytope

## 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