Unsolved Problems

Problem Given a Matrix $A$, the $k$-th elementary symmetric function of $A$, namely $S_k(A)$, is defined as the sum of all $k$-by- $k$ principal mino...

Problem Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?...

A latin square is even if the product of the signs of all of the row and column permutations is 1 and is odd otherwise. Conjecture For every positive...

Given integers $k,n\ge2$, let $d(k,n)$ be the smallest integer $d\ge2$ such that the symmetric group $\frak S$ on the set of all words of length $n$ o...

Let $E$ be a non-empty finite set. Given a partition $\bf h$ of $E$, the stabilizer of $\bf h$, denoted $S(\bf h)$, is the group formed by all permuta...

Let $S_n$ denote the set of all permutations of $[n]=\set{1,2,\ldots,n}$. Let $i(\pi)$ and $d(\pi)$ denote respectively the number of increasing and t...

Conjecture If ${\mathbb F}$ is a finite field with at least 4 elements and $A$ is an invertible $n \times n$ matrix with entries in ${\mathbb F}$, the...

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 spa...

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 unim...

Let $G$ be an (additive) abelian group, and for every $S \subseteq G$ let ${\mathit stab}(S) = \{ g \in G: g + S = S \}$. Conjecture Let $M$ be a mat...

Conjecture If $M_1,\ldots,M_k$ are matroids on $E$ and $\sum_{i=1}^k rk_{M_i}(X_i) \ge \ell (k-1)$ for every partition $\{X_1,\ldots,X_k\}$ of $E$, th...

Conjecture Let $r \ge 2$ be an integer and let $H$ be a minor minimal clutter with $\frac{1}{r}\tau_r(H) < \tau(H)$. Then either $H$ has a $J_k$ minor...

Conjecture If $A$ is 2-large, then $A$ is large....

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

Question What is the minimum cardinality of a decomposition of the $n$-cube into $n$-simplices?...

Conjecture If a finite set of unit balls in $\mathbb{R}^n$ is rearranged so that the distance between each pair of centers does not decrease, then the...

Conjecture Every set of $2^{n-2} + 1$ points in the plane in general position contains a subset of $n$ points which form a convex $n$-gon....

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 conve...

Question Is is possible to pack $n^n$ rectangular $n$-dimensional boxes each of which has side lengths $a_1,a_2,\ldots,a_n$ inside an $n$-dimensional ...

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 ...

Conjecture For every positive integer $k$, there exists an integer $d$ so that every polytope of dimension $\ge d$ has a $k$-dimensional face which is...

Conjecture Every convex polytope has a non-overlapping edge unfolding....

We let $p(G)$ denote the pebbling number of a graph $G$. Conjecture $p(G_1 \Box G_2) \le p(G_1) p(G_2)$....

Conjecture Every simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs...

Problem What is the Shannon capacity of $C_7$?...

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 o...

Problem ( $\dag$ ) Find a sufficient condition for a straight $\ell$-stage graph to be rearrangeable. In particular, what about a straight uniform gra...

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 connecti...

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

Conjecture (3-Decomposition Conjecture) Every connected cubic graph $G$ has a decomposition into a spanning tree, a family of cycles and a matching....

Conjecture Let $G$ be a $2$-connected cubic graph and let $S$ be a $2$-regular subgraph such that $G-E(S)$ is connected. Then $G$ has a cycle double c...

Conjecture For which values of $n$ and $d$ are there bi-colored graphs on $n$ vertices and $d$ different colors with the property that all the $d$ mon...

Conjecture For any bipartite graph $H$ and graph $G$, the number of homomorphisms from $H$ to $G$ is at least $\left(\frac{2|E(G)|}{|V(G)|^2}\right)^{...

Conjecture Suppose $G$ with $|V(G)|>2$ is a connected cubic graph admitting a $3$-edge coloring. Then there is an edge $e \in E(G)$ such that the cubi...

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

Problem Does every connected vertex-transitive graph have a Hamiltonian path?...

Problem Is there an eighth triangle free strongly regular graph?...

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 ${\ma...

Question Does every strongly regular graph have either itself or a complete graph as a core?...

Conjecture For every $\epsilon > 0$ and every positive integer $k$, there exists $r_0 = r_0(\epsilon,k)$ so that every simple $r$-regular graph $G$ wi...

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

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

Conjecture If $G = (V,E)$ is a graph, $p: E \rightarrow {\mathbb Z}$ is admissable, and $p(e)$ is even for every $e \in E(G)$, then $(G,p)$ has a fait...

Conjecture If $G$ is a 6-edge-connected Eulerian graph and $P$ is a 2-transition system for $G$, then $(G,P)$ has a compaible decomposition....

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

Conjecture If $G$ is a finite $r$-regular graph, where $r > 2$, then $G$ is not uniquely hamiltonian....

Conjecture Every 4-connected line graph is hamiltonian....

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

Question Is every Cayley graph Hamiltonian?...

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