# Random

## Fractional Hadwiger ★★

Author(s): Harvey; Reed; Seymour; Wood

**Conjecture**For every graph ,

(a)

(b)

(c) .

Keywords: fractional coloring, minors

## 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

## Inequality for square summable complex series ★★

Author(s): Retkes

**Conjecture**For all the following inequality holds

Keywords: Inequality

## Three-chromatic (0,2)-graphs ★★

Author(s): Payan

**Question**Are there any (0,2)-graphs with chromatic number exactly three?

Keywords:

## Funcoidal products inside an inward reloid ★★

Author(s): Porton

**Conjecture**(solved) If then for every funcoid and atomic f.o. and on the source and destination of correspondingly.

A stronger conjecture:

**Conjecture**If then for every funcoid and , .

Keywords: inward reloid

## 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

## 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

## Exponential Algorithms for Knapsack ★★

Author(s): Lipton

**Conjecture**

The famous 0-1 Knapsack problem is: Given and integers, determine whether or not there are values so that The best known worst-case algorithm runs in time times a polynomial in . Is there an algorithm that runs in time ?

Keywords: Algorithm construction; Exponential-time algorithm; Knapsack

## Triangle-packing vs triangle edge-transversal. ★★

Author(s): Tuza

**Conjecture**If has at most edge-disjoint triangles, then there is a set of edges whose deletion destroys every triangle.

Keywords:

## 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

## Directed path of length twice the minimum outdegree ★★★

Author(s): Thomassé

**Conjecture**Every oriented graph with minimum outdegree contains a directed path of length .

Keywords:

## Cube-Simplex conjecture ★★★

Author(s): Kalai

**Conjecture**For every positive integer , there exists an integer so that every polytope of dimension has a -dimensional face which is either a simplex or is combinatorially isomorphic to a -dimensional cube.

## 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

## 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

## Edge-antipodal colorings of cubes ★★

Author(s): Norine

We let denote the -dimensional cube graph. A map is called *edge-antipodal* if whenever are antipodal edges.

**Conjecture**If and is edge-antipodal, then there exist a pair of antipodal vertices which are joined by a monochromatic path.

Keywords: antipodal; cube; edge-coloring

## Growth of finitely presented groups ★★★

Author(s): Adyan

**Problem**Does there exist a finitely presented group of intermediate growth?

Keywords: finitely presented; growth

## Circular choosability of planar graphs ★

Author(s): Mohar

Let be a graph. If and are two integers, a -colouring of is a function from to such that for each edge . Given a list assignment of , i.e.~a mapping that assigns to every vertex a set of non-negative integers, an -colouring of is a mapping such that for every . A list assignment is a --list-assignment if and for each vertex . Given such a list assignment , the graph G is --colourable if there exists a --colouring , i.e. is both a -colouring and an -colouring. For any real number , the graph is --choosable if it is --colourable for every --list-assignment . Last, is circularly -choosable if it is --choosable for any , . The circular choosability (or circular list chromatic number or circular choice number) of G is

**Problem**What is the best upper bound on circular choosability for planar graphs?

Keywords: choosability; circular colouring; planar graphs

## 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:

## Woodall's Conjecture ★★★

Author(s): Woodall

**Conjecture**If is a directed graph with smallest directed cut of size , then has disjoint dijoins.

## Hamiltonian cycles in line graphs of infinite graphs ★★

Author(s): Georgakopoulos

**Conjecture**

- \item If is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. \item If the line graph of a locally finite graph is 4-connected, then is hamiltonian.

Keywords: hamiltonian; infinite graph; line graphs

## Linial-Berge path partition duality ★★★

**Conjecture**The minimum -norm of a path partition on a directed graph is no more than the maximal size of an induced -colorable subgraph.

Keywords: coloring; directed path; partition

## 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.

## Subset-sums equality (pigeonhole version) ★★★

Author(s):

**Problem**Let be natural numbers with . It follows from the pigeon-hole principle that there exist distinct subsets with . Is it possible to find such a pair in polynomial time?

Keywords: polynomial algorithm; search problem

## A homomorphism problem for flows ★★

Author(s): DeVos

**Conjecture**Let be abelian groups and let and satisfy and . If there is a homomorphism from to , then every graph with a B-flow has a B'-flow.

Keywords: homomorphism; nowhere-zero flow; tension

## Giuga's Conjecture on Primality ★★

Author(s): Giuseppe Giuga

**Conjecture**is a prime iff

Keywords: primality

## The Double Cap Conjecture ★★

Author(s): Kalai

**Conjecture**The largest measure of a Lebesgue measurable subset of the unit sphere of containing no pair of orthogonal vectors is attained by two open caps of geodesic radius around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

## 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:

## 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:

## Are almost all graphs determined by their spectrum? ★★★

Author(s):

**Problem**Are almost all graphs uniquely determined by the spectrum of their adjacency matrix?

Keywords: cospectral; graph invariant; spectrum

## Bounding the on-line choice number in terms of the choice number ★★

Author(s): Zhu

**Question**Are there graphs for which is arbitrarily large?

Keywords: choosability; list coloring; on-line choosability

## Algebraic independence of pi and e ★★★

Author(s):

**Conjecture**and are algebraically independent

Keywords: algebraic independence

## Are there only finite Fermat Primes? ★★★

Author(s):

**Conjecture**A Fermat prime is a Fermat number that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257 ,F_4=65537 It is unknown if other fermat primes exist.

Keywords:

## 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

## Antichains in the cycle continuous order ★★

Author(s): DeVos

If , are graphs, a function is called *cycle-continuous* if the pre-image of every element of the (binary) cycle space of is a member of the cycle space of .

**Problem**Does there exist an infinite set of graphs so that there is no cycle continuous mapping between and whenever ?

## Linear Hypergraphs with Dimension 3 ★★

Author(s): de Fraysseix; Ossona de Mendez; Rosenstiehl

**Conjecture**Any linear hypergraph with incidence poset of dimension at most 3 is the intersection hypergraph of a family of triangles and segments in the plane.

Keywords: Hypergraphs

## Packing T-joins ★★

Author(s): DeVos

**Conjecture**There exists a fixed constant (probably suffices) so that every graft with minimum -cut size at least contains a -join packing of size at least .

## Large acyclic induced subdigraph in a planar oriented graph. ★★

Author(s): Harutyunyan

**Conjecture**Every planar oriented graph has an acyclic induced subdigraph of order at least .

Keywords:

## 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

## 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:

## Inverse Galois Problem ★★★★

Author(s): Hilbert

**Conjecture**Every finite group is the Galois group of some finite algebraic extension of .

Keywords:

## Arc-disjoint directed cycles in regular directed graphs ★★

Author(s): Alon; McDiarmid; Molloy

**Conjecture**If is a -regular directed graph with no parallel arcs, then contains a collection of arc-disjoint directed cycles.

Keywords:

## Few subsequence sums in Z_n x Z_n ★★

**Conjecture**For every , the sequence in consisting of copes of and copies of has the fewest number of distinct subsequence sums over all zero-free sequences from of length .

Keywords: subsequence sum; zero sum

## Euler-Mascheroni constant ★★★

Author(s):

**Question**Is Euler-Mascheroni constant an transcendental number?

Keywords: constant; Euler; irrational; Mascheroni; rational; transcendental

## Oriented trees in n-chromatic digraphs ★★★

Author(s): Burr

**Conjecture**Every digraph with chromatic number at least contains every oriented tree of order as a subdigraph.

Keywords:

## Laplacian Degrees of a Graph ★★

Author(s): Guo

**Conjecture**If is a connected graph on vertices, then for .

Keywords: degree sequence; Laplacian matrix

## Erdös-Szekeres conjecture ★★★

**Conjecture**Every set of points in the plane in general position contains a subset of points which form a convex -gon.

Keywords: combinatorial geometry; Convex Polygons; ramsey theory

## The Berge-Fulkerson conjecture ★★★★

**Conjecture**If is a bridgeless cubic graph, then there exist 6 perfect matchings of with the property that every edge of is contained in exactly two of .

Keywords: cubic; perfect matching