Random

Number of Cliques in Minor-Closed Classes ★★

Author(s): Wood

Question   Is there a constant $ c $ such that every $ n $-vertex $ K_t $-minor-free graph has at most $ c^tn $ 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.

Keywords: complete geometric graph, edge colouring

Hoàng-Reed Conjecture ★★★

Author(s): Hoang; Reed

Conjecture   Every digraph in which each vertex has outdegree at least $ k $ contains $ k $ directed cycles $ C_1, \ldots, C_k $ such that $ C_j $ meets $ \cup_{i=1}^{j-1}C_i $ in at most one vertex, $ 2 \leq j \leq k $.

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?

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 $ L $ in $ S^3 $, let the symmetry group of $ L $ be denoted $ Sym(L) = \pi_0 Diff(S^3,L) $ ie: isotopy classes of diffeomorphisms of $ S^3 $ which preserve $ L $, where the isotopies are also required to preserve $ L $.

Now let $ L $ be a hyperbolic link. Assume $ L $ has the further `Brunnian' property that there exists a component $ L_0 $ of $ L $ such that $ L \setminus L_0 $ is the unlink. Let $ A_L $ be the subgroup of $ Sym(L) $ consisting of diffeomorphisms of $ S^3 $ which preserve $ L_0 $ together with its orientation, and which preserve the orientation of $ S^3 $.

There is a representation $ A_L \to \pi_0 Diff(L \setminus L_0) $ given by restricting the diffeomorphism to the $ L \setminus L_0 $. It's known that $ A_L $ is always a cyclic group. And $ \pi_0 Diff(L \setminus L_0) $ is a signed symmetric group -- the wreath product of a symmetric group with $ \mathbb Z_2 $.

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 $ G $? \[ \text{pair-cr}(G) = \text{cr}(G) \]

The crossing number $ \text{cr}(G) $ of a graph $ G $ is the minimum number of edge crossings in any drawing of $ G $ in the plane. In the pairwise crossing number $ \text{pair-cr}(G) $, 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.

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 $ k\geq 2 $, there is an integer $ f(k) $ so that every strongly $ f(k) $-connected tournament has $ k $ 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 \[ M_n  = 2^p  - 1 \] 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 $ G $ is a $ k $-chromatic graph on at most $ mk $ vertices, then $ \text{ch}(G)\leq \text{ch}(K_{m*k}) $.

Keywords: choosability; complete multipartite graph; list coloring

Forcing a 2-regular minor ★★

Author(s): Reed; Wood

Conjecture   Every graph with average degree at least $ \frac{4}{3}t-2 $ contains every 2-regular graph on $ t $ vertices as a minor.

Keywords: minors

Special Primes

Author(s): George BALAN

Conjecture   Let $ p $ be a prime natural number. Find all primes $ q\equiv1\left(\mathrm{mod}\: p\right) $, such that $ 2^{\frac{\left(q-1\right)}{p}}\equiv1\left(\mathrm{mod}\: q\right) $.

Keywords:

Long rainbow arithmetic progressions ★★

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

For $ k\in \mathbb{N} $ let $ T_k $ denote the minimal number $ t\in \mathbb{N} $ such that there is a rainbow $ AP(k) $ in every equinumerous $ t $-coloring of $ \{ 1,2,\ldots ,tn\} $ for every $ n\in \mathbb{N} $

Conjecture   For all $ k\geq 3 $, $ T_k=\Theta (k^2) $.

Keywords: arithmetic progression; rainbow

List chromatic number and maximum degree of bipartite graphs ★★

Author(s): Alon

Conjecture   There is a constant $ c $ such that the list chromatic number of any bipartite graph $ G $ of maximum degree $ \Delta $ is at most $ c \log \Delta $.

Keywords:

Decomposition of completions of reloids ★★

Author(s): Porton

Conjecture   For composable reloids $ f $ and $ g $ it holds
    \item $ \operatorname{Compl} ( g \circ f) = ( \operatorname{Compl} g) \circ f $ if $ f $ is a co-complete reloid; \item $ \operatorname{CoCompl} ( f \circ g) = f \circ \operatorname{CoCompl} g $ if $ f $ is a complete reloid; \item $ \operatorname{CoCompl} ( ( \operatorname{Compl} g) \circ f) = \operatorname{Compl} ( g \circ   ( \operatorname{CoCompl} f)) = ( \operatorname{Compl} g) \circ ( \operatorname{CoCompl} f) $; \item $ \operatorname{Compl} ( g \circ ( \operatorname{Compl} f)) = \operatorname{Compl} ( g \circ   f) $; \item $ \operatorname{CoCompl} ( ( \operatorname{CoCompl} g) \circ f) = \operatorname{CoCompl} ( g   \circ f) $.

Keywords: co-completion; completion; reloid

Hamilton cycle in small d-diregular graphs ★★

Author(s): Jackson

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

Conjecture   For $ d >2 $, every $ d $-diregular oriented graph on at most $ 4d+1 $ vertices has a Hamilton cycle.

Keywords:

Highly connected graphs with no K_n minor ★★★

Author(s): Thomas

Problem   Is it true for all $ n \ge 0 $, that every sufficiently large $ n $-connected graph without a $ K_n $ minor has a set of $ n-5 $ vertices whose deletion results in a planar graph?

Keywords: connectivity; minor

Lindelöf hypothesis ★★

Author(s): Lindelöf

Conjecture   For any $ \epsilon>0 $ $$\zeta\left(\frac12 + it\right) \mbox{ is }\mathcal{O}(t^\epsilon).$$

Since $ \epsilon $ can be replaced by a smaller value, we can also write the conjecture as, for any positive $ \epsilon $, $$\zeta\left(\frac12 + it\right) \mbox{ is }o(t^\varepsilon).$$

Keywords: Riemann Hypothesis; zeta

Weak saturation of the cube in the clique

Author(s): Morrison; Noel

Problem  

Determine $ \text{wsat}(K_n,Q_3) $.

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 $ f: o\rightarrow \{-, +\} $, where $ o $ is some ordinal. This $ o $ is the length of given sign expansion and also the birthday of the corresponding surreal number. Let us denote this length of $ s $ as $ \ell(s) $.

It is easy to prove that

$$ \ell(s+t) \leq \ell(s)+\ell(t) $$

What about

$$ \ell(s\times t) \leq \ell(s)\times\ell(t) $$

?

Keywords: surreal numbers

Ramsey properties of Cayley graphs ★★★

Author(s): Alon

Conjecture   There exists a fixed constant $ c $ so that every abelian group $ G $ has a subset $ S \subseteq G $ with $ -S = S $ so that the Cayley graph $ {\mathit Cayley}(G,S) $ has no clique or independent set of size $ > c \log |G| $.

Keywords: Cayley graph; Ramsey number

The Bermond-Thomassen Conjecture ★★

Author(s): Bermond; Thomassen

Conjecture   For every positive integer $ k $, every digraph with minimum out-degree at least $ 2k-1 $ contains $ k $ disjoint cycles.

Keywords: cycles

Sequence defined on multisets ★★

Author(s): Erickson

Conjecture   Define a $ 2 \times n $ 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 $ [1; 1] $, the sequence is: $ [1; 1] $ -> $ [1; 2] $ -> $ [1, 2; 1, 1] $ -> $ [1, 2; 3, 1] $ -> $ [1, 2, 3; 2, 1, 1] $ -> $ [1, 2, 3; 3, 2, 1] $ -> $ [1, 2, 3; 2, 2, 2] $ -> $ [1, 2, 3; 1, 4, 1] $ -> $ [1, 2, 3, 4; 3, 1, 1, 1] $ -> $ [1, 2, 3, 4; 4, 1, 2, 1] $ -> $ [1, 2, 3, 4; 3, 2, 1, 2] $ -> $ [1, 2, 3, 4; 2, 3, 2, 1] $, 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 $ G_1=(V,E) $ and its first arbitrary shortest spanning tree $ T_1=(V,E_1) $. We define the next graph $ G_2=(V,E\setminus E_1) $ and find on $ G_2 $ the second arbitrary shortest spanning tree $ T_2=(V,E_2) $. We continue similarly by finding $ T_3=(V,E_3) $ on $ G_3=(V,E\setminus \cup_{i=1}^{2}E_i) $, etc. Let k be the smallest number of disjoint shortest spanning trees as defined above and let $ T^{k}=(V,\cup_{i=1}^{k}E_i) $ be the graph obtained as union of all $ k $ disjoint trees.

Question 1. What is the smallest number of disjoint spanning trees creates a graph $ T^{k} $ containing a Hamiltonian path.

Question 2. What is the smallest number of disjoint spanning trees creates a graph $ T^{k} $ 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 $ {2^p-1} $ 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 ★★★

Author(s): Iradmusa

Conjecture   Let $ G $ be a graph and $ k $ be a positive integer. The $ k- $power of $ G $, denoted by $ G^k $, is defined on the vertex set $ V(G) $, by connecting any two distinct vertices $ x $ and $ y $ with distance at most $ k $. In other words, $ E(G^k)=\{xy:1\leq d_G(x,y)\leq k\} $. Also $ k- $subdivision of $ G $, denoted by $ G^\frac{1}{k} $, is constructed by replacing each edge $ ij $ of $ G $ with a path of length $ k $. Note that for $ k=1 $, we have $ G^\frac{1}{1}=G^1=G $.
Now we can define the fractional power of a graph as follows:
Let $ G $ be a graph and $ m,n\in \mathbb{N} $. The graph $ G^{\frac{m}{n}} $ is defined by the $ m- $power of the $ n- $subdivision of $ G $. In other words $ G^{\frac{m}{n}}\isdef (G^{\frac{1}{n}})^m $.
Conjecture. Let $ G $ be a connected graph with $ \Delta(G)\geq3 $ and $ m $ be a positive integer greater than 1. Then for any positive integer $ n>m $, we have $ \chi(G^{\frac{m}{n}})=\omega(G^\frac{m}{n}) $.
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 $ K_5 $ concentrated on a single value?

Keywords: random lifts, coloring

List Total Colouring Conjecture ★★

Author(s): Borodin; Kostochka; Woodall

Conjecture   If $ G $ is the total graph of a multigraph, then $ \chi_\ell(G)=\chi(G) $.

Keywords: list coloring; Total coloring; total graphs

Distribution and upper bound of mimic numbers ★★

Author(s): Bhattacharyya

Problem  

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

Definition   For any positive integer $ \mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i} $ divisible by $ \mathcal{D} $, the mimic function, $ f(\mathcal{D} | \mathcal{N}) $, is given by,

$$ f(\mathcal{D} | \mathcal{N}) = \sum_{i=0}^{n}\mathcal{X}_{i}(\mathcal{M}-\mathcal{D})^{i} $$

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

Definition   The number $ m $ is defined to be the mimic number of any positive integer $ \mathcal{N} = \sum_{i=0}^{n}\mathcal{X}_{i}\mathcal{M}^{i} $, with respect to $ \mathcal{D} $, for the minimum value of which $ f^{m}(\mathcal{D} | \mathcal{N}) = \mathcal{D} $.

Given these two definitions and a positive integer $ \mathcal{D} $, find the distribution of mimic numbers of those numbers divisible by $ \mathcal{D} $.

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer $ \mathcal{D} $.

Keywords: Divisibility; mimic function; mimic number

Cyclic spanning subdigraph with small cyclomatic number ★★

Author(s): Bondy

Conjecture   Let $ D $ be a digraph all of whose strong components are nontrivial. Then $ D $ contains a cyclic spanning subdigraph with cyclomatic number at most $ \alpha(D) $.

Keywords:

Cores of Cayley graphs ★★★★★

Author(s): Samal

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

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 $ k $-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 $ v_0,e_1,v_1,\ldots,v_m $ so that the vertex $ v_i $ is either the head of both $ e_i $ and $ e_{i+1} $ or the tail of both $ e_i $ and $ e_{i+1} $ for every $ 1 \le i \le m-1 $. A digraph is universal if for every pair of edges $ e,f $, there is an alternating walk containing both $ e $ and $ f $

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 $ H $ be a hypergraph. A strongly maximal matching is a matching $ F \subseteq E(H) $ so that $ |F' \setminus F| \le |F \setminus F'| $ for every matching $ F' $. A strongly minimal cover is a (vertex) cover $ X \subseteq V(H) $ so that $ |X' \setminus X| \ge |X \setminus X'| $ for every cover $ X' $.

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

Keywords: cover; infinite graph; matching

Birch & Swinnerton-Dyer conjecture ★★★★

Author(s):

Conjecture   Let $ E/K $ be an elliptic curve over a number field $ K $. Then the order of the zeros of its $ L $-function, $ L(E, s) $, at $ s = 1 $ is the Mordell-Weil rank of $ E(K) $.

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 $ R $ (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 $ P $ is defined to be $ (f_1 + f_2)/(f_0 + f_3) $ where $ f_i $ is the number of faces of $ P $ of dimension $ i $.

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

Keywords: f-vector; polytope

What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let $ \mathfrak{A} $ be an indexed family of sets.

Products are $ \prod A $ for $ A \in \prod \mathfrak{A} $.

Hyperfuncoids are filters $ \mathfrak{F} \Gamma $ on the lattice $ \Gamma $ of all finite unions of products.

Problem   Is $ \bigcap^{\mathsf{\tmop{FCD}}} $ a bijection from hyperfuncoids $ \mathfrak{F} \Gamma $ to:
    \item prestaroids on $ \mathfrak{A} $; \item staroids on $ \mathfrak{A} $; \item completary staroids on $ \mathfrak{A} $?

If yes, is $ \operatorname{up}^{\Gamma} $ defining the inverse bijection? If not, characterize the image of the function $ \bigcap^{\mathsf{\tmop{FCD}}} $ defined on $ \mathfrak{F} \Gamma $.

Consider also the variant of this problem with the set $ \Gamma $ replaced with the set $ \Gamma^{\ast} $ of complements of elements of the set $ \Gamma $.

Keywords: hyperfuncoids; multidimensional

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

Author(s):

Conjecture  (BEC-conjecture)   If $ G_1 $ and $ G_2 $ are $ n $-vertex graphs and $ (\Delta(G_1) + 1) (\Delta(G_2) + 1) < n + 1 $, then $ G_1 $ and $ G_2 $ 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 $ G $ and for every vertex $ v \in V(G) $ a subset $ \ell(v) $ of $ \{1,2,3,4\} $, and decides if there exists a partition of $ V(G) $ into $ \{A_1,A_2,A_3,A_4\} $ so that $ v \in A_i $ only if $ i \in \ell(v) $ and so that $ A_1,A_2 $ are independent, $ A_4 $ is a clique, and there are no edges between $ A_1 $ and $ A_3 $?

Keywords: list partition; polynomial algorithm

Rota's unimodal conjecture ★★★

Author(s): Rota

Let $ M $ be a matroid of rank $ r $, and for $ 0 \le i \le r $ let $ w_i $ be the number of closed sets of rank $ i $.

Conjecture   $ w_0,w_1,\ldots,w_r $ is unimodal.
Conjecture   $ w_0,w_1,\ldots,w_r $ is log-concave.

Keywords: flat; log-concave; matroid