# Random

## Perfect cuboid ★★

Author(s):

Conjecture   Does a perfect cuboid exist?

Keywords:

## Counterexamples to the Baillie-PSW primality test ★★

Author(s):

Problem  (1)   Find a counterexample to Baillie-PSW primality test or prove that there is no one.
Problem  (2)   Find a composite or which divides both (see Fermat pseudoprime) and the Fibonacci number (see Lucas pseudoprime), or prove that there is no such .

Keywords:

## Gao's theorem for nonabelian groups ★★

Author(s): DeVos

For every finite multiplicative group , let () denote the smallest integer so that every sequence of elements of has a subsequence of length (length ) which has product equal to 1 in some order.

Conjecture   for every finite group .

Keywords: subsequence sum; zero sum

## 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

## Closing Lemma for Diffeomorphism (Dynamical Systems) ★★★★

Author(s): Charles Pugh

Conjecture   Let and . Then for any neighborhood there is such that is periodic point of

There is an analogous conjecture for flows ( vector fields . In the case of diffeos this was proved by Charles Pugh for . In the case of Flows this has been solved by Sushei Hayahshy for . But in the two cases the problem is wide open for

Keywords: Dynamics , Pertubation

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

Author(s): Turan

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

Conjecture

Keywords: complete bipartite graph; crossing number

## Roller Coaster permutations ★★★

Author(s): Ahmed; Snevily

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:

## A conjecture about direct product of funcoids ★★

Author(s): Porton

Conjecture   Let and are monovalued, entirely defined funcoids with . Then there exists a pointfree funcoid such that (for every filter on ) (The join operation is taken on the lattice of filters with reversed order.)

A positive solution of this problem may open a way to prove that some funcoids-related categories are cartesian closed.

Keywords: category theory; general topology

## 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

## Covering powers of cycles with equivalence subgraphs ★

Author(s):

Conjecture   Given and , the graph has equivalence covering number .

Keywords:

## Obstacle number of planar graphs ★

Author(s): Alpert; Koch; Laison

Does there exist a planar graph with obstacle number greater than 1? Is there some such that every planar graph has obstacle number at most ?

## Rota's basis conjecture ★★★

Author(s): Rota

Conjecture   Let be a vector space of dimension and let be bases. Then there exist disjoint transversals of each of which is a base.

Keywords: base; latin square; linear algebra; matroid; transversal

## Reed's omega, delta, and chi conjecture ★★★

Author(s): Reed

For a graph , we define to be the maximum degree, to be the size of the largest clique subgraph, and to be the chromatic number of .

Conjecture   for every graph .

Keywords: coloring

## Degenerate colorings of planar graphs ★★★

Author(s): Borodin

A graph is -degenerate if every subgraph of has a vertex of degree .

Conjecture   Every simple planar graph has a 5-coloring so that for , the union of any color classes induces a -degenerate graph.

Keywords: coloring; degenerate; planar

## Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

Conjecture   for principal funcoid and a set of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

## Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index of a graph is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.

Question   Is it true that for every (sub)cubic graph , we have ?

Keywords: edge coloring; star coloring

## A discrete iteration related to Pierce expansions ★★

Author(s): Shallit

Conjecture   Let be integers. Set and for . Eventually we have ; put .

Example: , since , , , , , , , .

Prove or disprove: .

Keywords: Pierce expansions

## Erdős–Straus conjecture ★★

Author(s): Erdos; Straus

Conjecture

For all , there exist positive integers , , such that .

Keywords: Egyptian fraction

## A generalization of Vizing's Theorem? ★★

Author(s): Rosenfeld

Conjecture   Let be a simple -uniform hypergraph, and assume that every set of points is contained in at most edges. Then there exists an -edge-coloring so that any two edges which share vertices have distinct colors.

Keywords: edge-coloring; hypergraph; Vizing

## 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

## Barnette's Conjecture ★★★

Author(s): Barnette

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

Keywords: bipartite; cubic; hamiltonian

## Melnikov's valency-variety problem ★

Author(s): Melnikov

Problem   The valency-variety of a graph is the number of different degrees in . Is the chromatic number of any graph with at least two vertices greater than

Keywords:

## 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

## Goldbach conjecture ★★★★

Author(s): Goldbach

Conjecture   Every even integer greater than 2 is the sum of two primes.

## 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?

## The 3n+1 conjecture ★★★

Author(s): Collatz

Conjecture   Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .

Keywords: integer sequence

## 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.

Keywords: lucky; prime; seive

## The Two Color Conjecture ★★

Author(s): Neumann-Lara

Conjecture   If is an orientation of a simple planar graph, then there is a partition of into so that the graph induced by is acyclic for .

Keywords: acyclic; digraph; planar

## Oriented chromatic number of planar graphs ★★

Author(s):

An oriented colouring of an oriented graph is assignment of colours to the vertices such that no two arcs receive ordered pairs of colours and . It is equivalent to a homomorphism of the digraph onto some tournament of order .

Problem   What is the maximal possible oriented chromatic number of an oriented planar graph?

Keywords: oriented coloring; oriented graph; planar graph

## Which outer reloids are equal to inner ones ★★

Author(s): Porton

Warning: This formulation is vague (not exact).

Question   Characterize the set . In other words, simplify this formula.

The problem seems rather difficult.

Keywords:

## 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:

## The intersection of two perfect matchings ★★

Author(s): Macajova; Skoviera

Conjecture   Every bridgeless cubic graph has two perfect matchings , so that does not contain an odd edge-cut.

Keywords: cubic; nowhere-zero flow; perfect matching

## Good Edge Labelings ★★

Author(s): Araújo; Cohen; Giroire; Havet

Question   What is the maximum edge density of a graph which has a good edge labeling?

We say that a graph is good-edge-labeling critical, if it has no good edge labeling, but every proper subgraph has a good edge labeling.

Conjecture   For every , there is only a finite number of good-edge-labeling critical graphs with average degree less than .

Keywords: good edge labeling, edge labeling

## trace inequality ★★

Author(s):

Let be positive semidefinite, by Jensen's inequality, it is easy to see , whenever .

What about the , is it still valid?

Keywords:

## Inequality for square summable complex series ★★

Author(s): Retkes

Conjecture   For all the following inequality holds

Keywords: Inequality

## Random stable roommates ★★

Author(s): Mertens

Conjecture   The probability that a random instance of the stable roommates problem on people admits a solution is .

Keywords: stable marriage; stable roommates

## Decomposing a connected graph into paths. ★★★

Author(s): Gallai

Conjecture   Every simple connected graph on vertices can be decomposed into at most paths.

Keywords:

## Which lattices occur as intervals in subgroup lattices of finite groups? ★★★★

Author(s):

Conjecture

There exists a finite lattice that is not an interval in the subgroup lattice of a finite group.

Keywords: congruence lattice; finite groups

## Circular coloring triangle-free subcubic planar graphs ★★

Author(s): Ghebleh; Zhu

Problem   Does every triangle-free planar graph of maximum degree three have circular chromatic number at most ?

Keywords: circular coloring; planar graph; triangle free

## Graceful Tree Conjecture ★★★

Author(s):

Conjecture   All trees are graceful

Keywords: combinatorics; graceful labeling

## Combinatorial covering designs ★

Author(s): Gordon; Mills; Rödl; Schönheim

A covering design, or covering, is a family of -subsets, called blocks, chosen from a -set, such that each -subset is contained in at least one of the blocks. The number of blocks is the covering’s size, and the minimum size of such a covering is denoted by .

Problem   Find a closed form, recurrence, or better bounds for . Find a procedure for constructing minimal coverings.

Keywords: recreational mathematics

## Forcing a $K_6$-minor ★★

Author(s): Barát ; Joret; Wood

Conjecture   Every graph with minimum degree at least 7 contains a -minor.
Conjecture   Every 7-connected graph contains a -minor.

Keywords: connectivity; graph minors

## Sets with distinct subset sums ★★★

Author(s): Erdos

Say that a set has distinct subset sums if distinct subsets of have distinct sums.

Conjecture   There exists a fixed constant so that whenever has distinct subset sums.

Keywords: subset sum

## Infinite uniquely hamiltonian graphs ★★

Author(s): Mohar

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

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

Author(s):

Conjecture   Are all Mersenne Numbers with prime exponent Square free?

Keywords: Mersenne number

## 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

## 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:

## 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.

## Strong 5-cycle double cover conjecture ★★★

Author(s): Arthur; Hoffmann-Ostenhof

Conjecture   Let be a circuit in a bridgeless cubic graph . Then there is a five cycle double cover of such that is a subgraph of one of these five cycles.

Keywords: cycle cover

## Hamilton decomposition of prisms over 3-connected cubic planar graphs ★★

Author(s): Alspach; Rosenfeld

Conjecture   Every prism over a -connected cubic planar graph can be decomposed into two Hamilton cycles.

Keywords: