# Random

## Polignac's Conjecture ★★★

Author(s): de Polignac

**Conjecture**Polignac's Conjecture: For any positive even number n, there are infinitely many prime gaps of size n. In other words: There are infinitely many cases of two consecutive prime numbers with difference n.

In particular, this implies:

**Conjecture**Twin Prime Conjecture: There are an infinite number of twin primes.

## 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:

## 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:

## Coloring and immersion ★★★

Author(s): Abu-Khzam; Langston

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

Keywords: coloring; complete graph; immersion

## Wide partition conjecture ★★

**Conjecture**An integer partition is wide if and only if it is Latin.

Keywords:

## Does the chromatic symmetric function distinguish between trees? ★★

Author(s): Stanley

**Problem**Do there exist non-isomorphic trees which have the same chromatic symmetric function?

Keywords: chromatic polynomial; symmetric function; tree

## Consecutive non-orientable embedding obstructions ★★★

Author(s):

**Conjecture**Is there a graph that is a minor-minimal obstruction for two non-orientable surfaces?

## Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

**Conjecture**If , then contains at least cycles of length .

Keywords: chromatic number; cycles

## (m,n)-cycle covers ★★★

Author(s): Celmins; Preissmann

**Conjecture**Every bridgeless graph has a (5,2)-cycle-cover.

## Frobenius number of four or more integers ★★

Author(s):

**Problem**Find an explicit formula for Frobenius number of co-prime positive integers for .

Keywords:

## Minimal graphs with a prescribed number of spanning trees ★★

Author(s): Azarija; Skrekovski

**Conjecture**Let be an integer and let denote the least integer such that there exists a simple graph on vertices having precisely spanning trees. Then

Keywords: number of spanning trees, asymptotics

## 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:

## Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

Let be a positive integer. We say that a graph is *strongly -colorable* if for every partition of the vertices to sets of size at most there is a proper -coloring of in which the vertices in each set of the partition have distinct colors.

**Conjecture**If is the maximal degree of a graph , then is strongly -colorable.

Keywords: strong coloring

## 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

## Double-critical graph conjecture ★★

A connected simple graph is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two.

**Conjecture**is the only -chromatic double-critical graph

Keywords: coloring; complete graph

## 4-connected graphs are not uniquely hamiltonian ★★

Author(s): Fleischner

**Conjecture**Every -connected graph with a Hamilton cycle has a second Hamilton cycle.

Keywords:

## 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:

## 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 ?

Keywords: graph drawing; obstacle number; planar graph; visibility graph

## Fractional Hadwiger ★★

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

**Conjecture**For every graph ,

(a)

(b)

(c) .

Keywords: fractional coloring, minors

## 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:

## Covering powers of cycles with equivalence subgraphs ★

Author(s):

**Conjecture**Given and , the graph has equivalence covering number .

Keywords:

## 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

## 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

## 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

## 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

## Acyclic edge-colouring ★★

Author(s): Fiamcik

**Conjecture**Every simple graph with maximum degree has a proper -edge-colouring so that every cycle contains edges of at least three distinct colours.

Keywords: edge-coloring

## Are vertex minor closed classes chi-bounded? ★★

Author(s): Geelen

**Question**Is every proper vertex-minor closed class of graphs chi-bounded?

Keywords: chi-bounded; circle graph; coloring; vertex minor

## Diophantine quintuple conjecture ★★

Author(s):

**Definition**A set of m positive integers is called a Diophantine -tuple if is a perfect square for all .

**Conjecture (1)**Diophantine quintuple does not exist.

It would follow from the following stronger conjecture [Da]:

**Conjecture (2)**If is a Diophantine quadruple and , then

Keywords:

## Characterizing (aleph_0,aleph_1)-graphs ★★★

Call a graph an -*graph* if it has a bipartition so that every vertex in has degree and every vertex in has degree .

**Problem**Characterize the -graphs.

Keywords: binary tree; infinite graph; normal spanning tree; set theory

## Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems) ★★★★

Author(s):

**Conjecture**Let be the space of Diffeomorphisms on the connected , compact and boundaryles manifold M and the space of vector fields. There is a dense set ( ) such that exhibit a finite number of attractor whose basins cover Lebesgue almost all ambient space

This is a very Deep and Hard problem in Dynamical Systems . It present the dream of the dynamicist mathematicians .

Keywords: Attractors , basins, Finite

## inverse of an integer matrix ★★

Author(s): Gregory

**Question**I've been working on this for a long time and I'm getting nowhere. Could you help me or at least tell me where to look for help. Suppose D is an m-by-m diagonal matrix with integer elements all . Suppose X is an m-by-n integer matrix . Consider the partitioned matrix M = [D X]. Obviously M has full row rank so it has a right inverse of rational numbers. The question is, under what conditions does it have an integer right inverse? My guess, which I can't prove, is that the integers in each row need to be relatively prime.

Keywords: invertable matrices, integer matrices

## $C^r$ Stability Conjecture ★★★★

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

Keywords: diffeomorphisms,; dynamical systems

## Fundamental group torsion for subsets of Euclidean 3-space ★★

Author(s): Ancient/folklore

**Problem**Does there exist a subset of such that its fundamental group has an element of finite order?

Keywords: subsets of euclidean space; torsion

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

Author(s):

**Conjecture**

Skewes' number is not an integer.

Keywords:

## The Borodin-Kostochka Conjecture ★★

**Conjecture**Every graph with maximum degree has chromatic number at most .

Keywords:

## Counting 3-colorings of the hex lattice ★★

Author(s): Thomassen

**Problem**Find .

Keywords: coloring; Lieb's Ice Constant; tiling; torus

## 5-local-tensions ★★

Author(s): DeVos

**Conjecture**There exists a fixed constant (probably suffices) so that every embedded (loopless) graph with edge-width has a 5-local-tension.

## Partitionning a tournament into k-strongly connected subtournaments. ★★

Author(s): Thomassen

**Problem**Let be positve integer Does there exists an integer such that every -strong tournament admits a partition of its vertex set such that the subtournament induced by is a non-trivial -strong for all .

Keywords:

## Nonrepetitive colourings of planar graphs ★★

Author(s): Alon N.; Grytczuk J.; Hałuszczak M.; Riordan O.

**Question**Do planar graphs have bounded nonrepetitive chromatic number?

Keywords: nonrepetitive colouring; planar graphs

## Dirac's Conjecture ★★

Author(s): Dirac

**Conjecture**For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .

Keywords: point set

## 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

## 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

## Geodesic cycles and Tutte's Theorem ★★

Author(s): Georgakopoulos; Sprüssel

**Problem**If is a -connected finite graph, is there an assignment of lengths to the edges of , such that every -geodesic cycle is peripheral?

Keywords: cycle space; geodesic cycles; peripheral cycles

## Monadic second-order logic with cardinality predicates ★★

Author(s): Courcelle

The problem concerns the extension of Monadic Second Order Logic (over a binary relation representing the edge relation) with the following atomic formulas:

- \item \item

where is a fixed recursive set of integers.

Let us fix and a closed formula in this language.

**Conjecture**Is it true that the validity of for a graph of tree-width at most can be tested in polynomial time in the size of ?

Keywords: bounded tree width; cardinality predicates; FMT03-Bedlewo; MSO

## Bounding the chromatic number of triangle-free graphs with fixed maximum degree ★★

**Conjecture**A triangle-free graph with maximum degree has chromatic number at most .

Keywords: chromatic number; girth; maximum degree; triangle free

## Algorithm for graph homomorphisms ★★

Author(s): Fomin; Heggernes; Kratsch

**Question**

Is there an algorithm that decides, for input graphs and , whether there exists a homomorphism from to in time for some constant ?

Keywords: algorithm; Exponential-time algorithm; homomorphism

## Ryser's conjecture ★★★

Author(s): Ryser

**Conjecture**Let be an -uniform -partite hypergraph. If is the maximum number of pairwise disjoint edges in , and is the size of the smallest set of vertices which meets every edge, then .

Keywords: hypergraph; matching; packing

## Slice-ribbon problem ★★★★

Author(s): Fox

**Conjecture**Given a knot in which is slice, is it a ribbon knot?

## 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

## Finding k-edge-outerplanar graph embeddings ★★

Author(s): Bentz

**Conjecture**It has been shown that a -outerplanar embedding for which is minimal can be found in polynomial time. Does a similar result hold for -edge-outerplanar graphs?

Keywords: planar graph; polynomial algorithm