Chords of longest cycles ★★★

Author(s): Thomassen

Conjecture   If $ G $ is a 3-connected graph, every longest cycle in $ G $ has a chord.

Keywords: chord; connectivity; cycle

Edge-Unfolding Convex Polyhedra ★★

Author(s): Shephard

Conjecture   Every convex polyhedron has a (nonoverlapping) edge unfolding.

Keywords: folding; nets

Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index $ \chi_s'(G) $ of a graph $ G $ 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 $ G $, we have $ \chi_s'(G) \le 6 $?

Keywords: edge coloring; star coloring

Upgrading a completary multifuncoid ★★

Author(s): Porton

Let $ \mho $ be a set, $ \mathfrak{F} $ be the set of filters on $ \mho $ ordered reverse to set-theoretic inclusion, $ \mathfrak{P} $ be the set of principal filters on $ \mho $, let $ n $ be an index set. Consider the filtrator $ \left( \mathfrak{F}^n ; \mathfrak{P}^n \right) $.

Conjecture   If $ f $ is a completary multifuncoid of the form $ \mathfrak{P}^n $, then $ E^{\ast} f $ is a completary multifuncoid of the form $ \mathfrak{F}^n $.

See below for definition of all concepts and symbols used to in this conjecture.

Refer to this Web site for the theory which I now attempt to generalize.


2-accessibility of primes ★★

Author(s): Landman; Robertson

Question   Is the set of prime numbers 2-accessible?

Keywords: monochromatic diffsequences; primes

Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let $ \delta $ be a proximity.

A set $ A $ is connected regarding $ \delta $ iff $ \forall X,Y \in \mathscr{P} A \setminus \{ \emptyset \} : \left( X \cup Y = A \Rightarrow X \mathrel{\delta} Y \right) $.

Conjecture   The following statements are equivalent for every endofuncoid $ \mu $ and a set $ U $:
    \item $ U $ is connected regarding $ \mu $. \item For every $ a, b \in U $ there exists a totally ordered set $ P \subseteq   U $ such that $ \min P = a $, $ \max P = b $, and for every partion $ \{ X, Y \} $ of $ P $ into two sets $ X $, $ Y $ such that $ \forall x \in X, y \in Y : x < y $, we have $ X \mathrel{[ \mu]^{\ast}} Y $.

Keywords: connected; connectedness; proximity space

Strong colorability ★★★

Author(s): Aharoni; Alon; Haxell

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

Conjecture   If $ \Delta $ is the maximal degree of a graph $ G $, then $ G $ is strongly $ 2 \Delta $-colorable.

Keywords: strong coloring

3-accessibility of Fibonacci numbers ★★

Author(s): Landman; Robertson

Question   Is the set of Fibonacci numbers 3-accessible?

Keywords: Fibonacci numbers; monochromatic diffsequences

Decomposing an even tournament in directed paths. ★★★

Author(s): Alspach; Mason; Pullman

Conjecture   Every tournament $ D $ on an even number of vertices can be decomposed into $ \sum_{v\in V}\max\{0,d^+(v)-d^-(v)\} $ directed paths.


Refuting random 3SAT-instances on $O(n)$ clauses (weak form) ★★★

Author(s): Feige

Conjecture   For every rational $ \epsilon > 0 $ and every rational $ \Delta $, there is no polynomial-time algorithm for the following problem.

Given is a 3SAT (3CNF) formula $ I $ on $ n $ variables, for some $ n $, and $ m = \floor{\Delta n} $ clauses drawn uniformly at random from the set of formulas on $ n $ variables. Return with probability at least 0.5 (over the instances) that $ I $ is typical without returning typical for any instance with at least $ (1 - \epsilon)m $ simultaneously satisfiable clauses.

Keywords: NP; randomness in TCS; satisfiability

Extremal problem on the number of tree endomorphism ★★

Author(s): Zhicong Lin

Conjecture   An endomorphism of a graph is a mapping on the vertex set of the graph which preserves edges. Among all the $ n $ vertices' trees, the star with $ n $ vertices has the most endomorphisms, while the path with $ n $ vertices has the least endomorphisms.


Every metamonovalued reloid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued reloid is monovalued.


Jones' conjecture ★★

Author(s): Kloks; Lee; Liu

For a graph $ G $, let $ cp(G) $ denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let $ cc(G) $ denote the cardinality of a minimum feedback vertex set (set of vertices $ X $ so that $ G-X $ is acyclic).

Conjecture   For every planar graph $ G $, $ cc(G)\leq 2cp(G) $.

Keywords: cycle packing; feedback vertex set; planar graph

Bouchet's 6-flow conjecture ★★★

Author(s): Bouchet

Conjecture   Every bidirected graph with a nowhere-zero $ k $-flow for some $ k $, has a nowhere-zero $ 6 $-flow.

Keywords: bidirected graph; nowhere-zero flow

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 $ \ge 2 $. Suppose X is an m-by-n integer matrix $ (m \le n) $. 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

Subdivision of a transitive tournament in digraphs with large outdegree. ★★

Author(s): Mader

Conjecture   For all $ k $ there is an integer $ f(k) $ such that every digraph of minimum outdegree at least $ f(k) $ contains a subdivision of a transitive tournament of order $ k $.


57-regular Moore graph? ★★★

Author(s): Hoffman; Singleton

Question   Does there exist a 57-regular graph with diameter 2 and girth 5?

Keywords: cage; Moore graph

Petersen coloring conjecture ★★★

Author(s): Jaeger

Conjecture   Let $ G $ be a cubic graph with no bridge. Then there is a coloring of the edges of $ G $ using the edges of the Petersen graph so that any three mutually adjacent edges of $ G $ map to three mutually adjancent edges in the Petersen graph.

Keywords: cubic; edge-coloring; Petersen graph

Do any three longest paths in a connected graph have a vertex in common? ★★

Author(s): Gallai

Conjecture   Do any three longest paths in a connected graph have a vertex in common?


Linear-size circuits for stable $0,1 < 2$ sorting? ★★

Author(s): Regan

Problem   Can $ O(n) $-size circuits compute the function $ f $ on $ \{0,1,2\}^* $ defined inductively by $ f(\lambda) = \lambda $, $ f(0x) = 0f(x) $, $ f(1x) = 1f(x) $, and $ f(2x) = f(x)2 $?

Keywords: Circuits; sorting

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

The Double Cap Conjecture ★★

Author(s): Kalai

Conjecture   The largest measure of a Lebesgue measurable subset of the unit sphere of $ \mathbb{R}^n $ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $ \pi/4 $ around the north and south poles.

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

Degenerate colorings of planar graphs ★★★

Author(s): Borodin

A graph $ G $ is $ k $-degenerate if every subgraph of $ G $ has a vertex of degree $ \le k $.

Conjecture   Every simple planar graph has a 5-coloring so that for $ 1 \le k \le 4 $, the union of any $ k $ color classes induces a $ (k-1) $-degenerate graph.

Keywords: coloring; degenerate; planar

Olson's Conjecture ★★

Author(s): Olson

Conjecture   If $ a_1,a_2,\ldots,a_{2n-1} $ is a sequence of elements from a multiplicative group of order $ n $, then there exist $ 1 \le j_1 < j_2 \ldots < j_n \le 2n-1 $ so that $ \prod_{i=1}^n a_{j_i} = 1 $.

Keywords: zero sum

Kriesell's Conjecture ★★

Author(s): Kriesell

Conjecture   Let $ G $ be a graph and let $ T\subseteq V(G) $ such that for any pair $ u,v\in T $ there are $ 2k $ edge-disjoint paths from $ u $ to $ v $ in $ G $. Then $ G $ contains $ k $ edge-disjoint trees, each of which contains $ T $.

Keywords: Disjoint paths; edge-connectivity; spanning trees

Big Line or Big Clique in Planar Point Sets ★★

Author(s): Kara; Por; Wood

Let $ S $ be a set of points in the plane. Two points $ v $ and $ w $ in $ S $ are visible with respect to $ S $ if the line segment between $ v $ and $ w $ contains no other point in $ S $.

Conjecture   For all integers $ k,\ell\geq2 $ there is an integer $ n $ such that every set of at least $ n $ points in the plane contains at least $ \ell $ collinear points or $ k $ pairwise visible points.

Keywords: Discrete Geometry; Geometric Ramsey Theory

Jacobian Conjecture ★★★

Author(s): Keller

Conjecture   Let $ k $ be a field of characteristic zero. A collection $ f_1,\ldots,f_n $ of polynomials in variables $ x_1,\ldots,x_n $ defines an automorphism of $ k^n $ if and only if the Jacobian matrix is a nonzero constant.

Keywords: Affine Geometry; Automorphisms; Polynomials

The additive basis conjecture ★★★

Author(s): Jaeger; Linial; Payan; Tarsi

Conjecture   For every prime $ p $, there is a constant $ c(p) $ (possibly $ c(p)=p $) so that the union (as multisets) of any $ c(p) $ bases of the vector space $ ({\mathbb Z}_p)^n $ contains an additive basis.

Keywords: additive basis; matrix

Graph product of multifuncoids ★★

Author(s): Porton

Conjecture   Let $ F $ is a family of multifuncoids such that each $ F_i $ is of the form $ \lambda j \in N \left( i \right) : \mathfrak{F} \left( U_j \right) $ where $ N \left( i \right) $ is an index set for every $ i $ and $ U_j $ is a set for every $ j $. Let every $ F_i = E^{\ast} f_i $ for some multifuncoid $ f_i $ of the form $ \lambda j \in N \left( i \right) : \mathfrak{P} \left( U_j \right) $ regarding the filtrator $ \left( \prod_{j \in N \left( i \right)} \mathfrak{F} \left( U_j \right) ; \prod_{j \in N \left( i \right)} \mathfrak{P} \left( U_j \right) \right) $. Let $ H $ is a graph-composition of $ F $ (regarding some partition $ G $ and external set $ Z $). Then there exist a multifuncoid $ h $ of the form $ \lambda j \in Z : \mathfrak{P} \left( U_j \right) $ such that $ H = E^{\ast} h $ regarding the filtrator $ \left( \prod_{j \in Z} \mathfrak{F} \left( U_j \right) ; \prod_{j \in Z} \mathfrak{P} \left( U_j \right) \right) $.

Keywords: graph-product; multifuncoid

Jaeger's modular orientation conjecture ★★★

Author(s): Jaeger

Conjecture   Every $ 4k $-edge-connected graph can be oriented so that $ {\mathit indegree}(v) - {\mathit outdegree}(v) \cong 0 $ (mod $ 2k+1 $) for every vertex $ v $.

Keywords: nowhere-zero flow; orientation

Arc-disjoint out-branching and in-branching ★★

Author(s): Thomassen

Conjecture   There exists an integer $ k $ such that every $ k $-arc-strong digraph $ D $ with specified vertices $ u $ and $ v $ contains an out-branching rooted at $ u $ and an in-branching rooted at $ v $ which are arc-disjoint.


Woodall's Conjecture ★★★

Author(s): Woodall

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

Keywords: digraph; packing

Waring rank of determinant ★★

Author(s): Teitler

Question   What is the Waring rank of the determinant of a $ d \times d $ generic matrix?

For simplicity say we work over the complex numbers. The $ d \times d $ generic matrix is the matrix with entries $ x_{i,j} $ for $ 1 \leq i,j \leq d $. Its determinant is a homogeneous form of degree $ d $, in $ d^2 $ variables. If $ F $ is a homogeneous form of degree $ d $, a power sum expression for $ F $ is an expression of the form $ F = \ell_1^d+\dotsb+\ell_r^d $, the $ \ell_i $ (homogeneous) linear forms. The Waring rank of $ F $ is the least number of terms $ r $ in any power sum expression for $ F $. For example, the expression $ xy = \frac{1}{4}(x+y)^2 - \frac{1}{4}(x-y)^2 $ means that $ xy $ has Waring rank $ 2 $ (it can't be less than $ 2 $, as $ xy \neq \ell_1^2 $).

The $ 2 \times 2 $ generic determinant $ x_{1,1}x_{2,2}-x_{1,2}x_{2,1} $ (or $ ad-bc $) has Waring rank $ 4 $. The Waring rank of the $ 3 \times 3 $ generic determinant is at least $ 14 $ and no more than $ 20 $, see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.

Keywords: Waring rank, determinant

Concavity of van der Waerden numbers ★★

Author(s): Landman

For $ k $ and $ \ell $ positive integers, the (mixed) van der Waerden number $ w(k,\ell) $ is the least positive integer $ n $ such that every (red-blue)-coloring of $ [1,n] $ admits either a $ k $-term red arithmetic progression or an $ \ell $-term blue arithmetic progression.

Conjecture   For all $ k $ and $ \ell $ with $ k \geq \ell $, $ w(k,\ell) \geq w(k+1,\ell-1) $.

Keywords: arithmetic progression; van der Waerden

Shuffle-Exchange Conjecture (graph-theoretic form) ★★★

Author(s): Beneš; Folklore; Stone

Given integers $ k,n \ge 2 $, the 2-stage Shuffle-Exchange graph/network, denoted $ \text{SE}(k,n) $, is the simple $ k $-regular bipartite graph with the ordered pair $ (U,V) $ of linearly labeled parts $ U:=\{u_0,\dots,u_{t-1}\} $ and $ V:=\{v_0,\dots,v_{t-1}\} $, where $ t:=k^{n-1} $, such that vertices $ u_i $ and $ v_j $ are adjacent if and only if $ (j - ki) \text{ mod } t < k $ (see Fig.1).

Given integers $ k,n,r \ge 2 $, the $ r $-stage Shuffle-Exchange graph/network, denoted $ (\text{SE}(k,n))^{r-1} $, is the proper (i.e., respecting all the orders) concatenation of $ r-1 $ identical copies of $ \text{SE}(k,n) $ (see Fig.1).

Let $ r(k,n) $ be the smallest integer $ r\ge 2 $ such that the graph $ (\text{SE}(k,n))^{r-1} $ is rearrangeable.

Problem   Find $ r(k,n) $.
Conjecture   $ r(k,n)=2n-1 $.


Graphs of exact colorings ★★


Conjecture For $  c \geq m \geq 1  $, let $  P(c,m)  $ be the statement that given any exact $  c  $-coloring of the edges of a complete countably infinite graph (that is, a coloring with $  c  $ colors all of which must be used at least once), there exists an exactly $  m  $-colored countably infinite complete subgraph. Then $  P(c,m)  $ is true if and only if $  m=1  $, $  m=2  $, or $  c=m  $.


Gao's theorem for nonabelian groups ★★

Author(s): DeVos

For every finite multiplicative group $ G $, let $ s(G) $ ($ s'(G) $) denote the smallest integer $ m $ so that every sequence of $ m $ elements of $ G $ has a subsequence of length $ >0 $ (length $ |G| $) which has product equal to 1 in some order.

Conjecture   $ s'(G) = s(G) + |G| - 1 $ for every finite group $ G $.

Keywords: subsequence sum; zero sum

Monochromatic empty triangles ★★★


If $ X \subseteq {\mathbb R}^2 $ is a finite set of points which is 2-colored, an empty triangle is a set $ T \subseteq X $ with $ |T|=3 $ so that the convex hull of $ T $ is disjoint from $ X \setminus T $. We say that $ T $ is monochromatic if all points in $ T $ are the same color.

Conjecture   There exists a fixed constant $ c $ with the following property. If $ X \subseteq {\mathbb R}^2 $ is a set of $ n $ points in general position which is 2-colored, then it has $ \ge cn^2 $ monochromatic empty triangles.

Keywords: empty triangle; general position; ramsey theory

$C^r$ Stability Conjecture ★★★★

Author(s): Palis; Smale

Conjecture   Any $ C^r $ structurally stable diffeomorphism is hyperbolic.

Keywords: diffeomorphisms,; dynamical systems

The circular embedding conjecture ★★★

Author(s): Haggard

Conjecture   Every 2-connected graph may be embedded in a surface so that the boundary of each face is a cycle.

Keywords: cover; cycle

Dense rational distance sets in the plane ★★★

Author(s): Ulam

Problem   Does there exist a dense set $ S \subseteq {\mathbb R}^2 $ so that all pairwise distances between points in $ S $ are rational?

Keywords: integral distance; rational distance

Graphs with a forbidden induced tree are chi-bounded ★★★

Author(s): Gyarfas

Say that a family $ {\mathcal F} $ of graphs is $ \chi $-bounded if there exists a function $ f: {\mathbb N} \rightarrow {\mathbb N} $ so that every $ G \in {\mathcal F} $ satisfies $ \chi(G) \le f (\omega(G)) $.

Conjecture   For every fixed tree $ T $, the family of graphs with no induced subgraph isomorphic to $ T $ is $ \chi $-bounded.

Keywords: chi-bounded; coloring; excluded subgraph; tree

Finding k-edge-outerplanar graph embeddings ★★

Author(s): Bentz

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

Keywords: planar graph; polynomial algorithm

Diagonal Ramsey numbers ★★★★

Author(s): Erdos

Let $ R(k,k) $ denote the $ k^{th} $ diagonal Ramsey number.

Conjecture   $ \lim_{k \rightarrow \infty} R(k,k) ^{\frac{1}{k}} $ exists.
Problem   Determine the limit in the above conjecture (assuming it exists).

Keywords: Ramsey number

Drawing disconnected graphs on surfaces ★★

Author(s): DeVos; Mohar; Samal

Conjecture   Let $ G $ be the disjoint union of the graphs $ G_1 $ and $ G_2 $ and let $ \Sigma $ be a surface. Is it true that every optimal drawing of $ G $ on $ \Sigma $ has the property that $ G_1 $ and $ G_2 $ are disjoint?

Keywords: crossing number; surface

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

Arc-disjoint directed cycles in regular directed graphs ★★

Author(s): Alon; McDiarmid; Molloy

Conjecture   If $ G $ is a $ k $-regular directed graph with no parallel arcs, then $ G $ contains a collection of $ {k+1 \choose 2} $ arc-disjoint directed cycles.


Primitive pythagorean n-tuple tree ★★


Conjecture   Find linear transformation construction of primitive pythagorean n-tuple tree!


Unions of triangle free graphs ★★★

Author(s): Erdos; Hajnal

Problem   Does there exist a graph with no subgraph isomorphic to $ K_4 $ which cannot be expressed as a union of $ \aleph_0 $ triangle free graphs?

Keywords: forbidden subgraph; infinite graph; triangle free

Stable set meeting all longest directed paths. ★★

Author(s): Laborde; Payan; Xuong N.H.

Conjecture   Every digraph has a stable set meeting all longest directed paths