Unsolved Problems

Conjecture Let $D$ be a digraph all of whose strong components are nontrivial. Then $D$ contains a cyclic spanning subdigraph with cyclomatic number a...

Conjecture Every planar oriented graph $D$ has an acyclic induced subdigraph of order at least $\frac{3}{5} |V(D)|$....

Conjecture Let $\ell \geq 2$ be an integer. For every integer $n\geq 0$, there exists an integer $t_n=t_n(\ell)$ such that for every digraph $D$, eith...

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....

Problem Let $k_1, \dots, k_p$ be positve integer Does there exists an integer $g(k_1, \dots, k_p)$ such that every $g(k_1, \dots, k_p)$-strong tournam...

Conjecture Every k-arc-strong tournament decomposes into k spanning strong digraphs....

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 ...

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 w...

An $H$-factor in a graph $G$ is a set of vertex-disjoint copies of $H$ covering all vertices of $G$. Problem Let $c$ be a fixed positive real number ...

Conjecture If $G$ is a simple triangle-free graph, then there is a set of at most $n^2/25$ edges whose deletion destroys every odd cycle....

Conjecture If $G$ has at most $k$ edge-disjoint triangles, then there is a set of $2k$ edges whose deletion destroys every triangle....

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

Setup Fix a tree $T$ and for every vertex $v \in V(T)$ a non-negative integer $g(v)$ which we think of as the amount of gold at $v$. 2-Player game Pl...

Question Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments?...

Conjecture Let $H$ be a $k$-uniform hypergraph. If $H$ is a critical $k$-forest, then it is a $k$-tree....

Conjecture Let $F$ be a finite family of finite sets, not all empty, that is closed under taking unions. Then there exists $x$ such that $x$ is an ele...

Problem Let $H_1$ and $H_2$ be two $r$-uniform hypergraph on the same vertex set $V$. Does there always exist a partition of $V$ into $r$ classes $V_1...

Conjecture Every simple $3$-uniform hypergraph on $3n$ vertices which contains no complete $3$-uniform hypergraph on four vertices has at most $\frac1...

Problem Are there any uniquely hamiltonian locally finite 1-ended graphs which are regular of degree $r > 2$?...

Conjecture - If $G$ is a 4-edge-connected locally finite graph, then its line graph is hamiltonian. - If the line graph $L(G)$ of a locally finite gr...

Conjecture - If $G$ is a countable connected graph then its third power is hamiltonian. - If $G$ is a 2-connected countable graph then its square is ...

Conjecture If $G$ is a highly arc transitive digraph with two ends, then every tile of $G$ is a disjoint union of complete bipartite graphs....

Problem Let $G$ be a graph, $\omega$ a countable end of $G$, and $K$ an infinite set of pairwise disjoint $\omega$-rays in $G$. Prove that there is a ...

If $G$ is a graph and $p \in [0,1]$, we let $G_p$ denote a subgraph of $G$ where each edge of $G$ appears in $G_p$ with independently with probability...

Conjecture Let $G$ be a finite graph, let $e,f \in E(G)$, and let $F$ be the edge set of a forest chosen uniformly at random from all forests of $G$. ...

Question Is the chromatic number of a random lift of $K_5$ concentrated on a single value?...

Conjecture Every sufficiently large plane triangulation $G$ has a dominating set of size $\le \frac{1}{4} |V(G)|$....

Conjecture Every planar graph on $n$ verices has an induced forest with at least $n/2$ vertices....

Conjecture Every 4-connected toroidal graph has a Hamilton cycle....

Conjecture There exists a fixed constant $c$ (probably $c=4$ suffices) so that every embedded (loopless) graph with edge-width $\ge c$ has a 5-local-t...

Consider a set $S$ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $S$ has a vertex for each intersect...

The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane. The $d$-dimensional (hyper)cube $Q_d$ is t...

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 ...

Conjecture Let $(a_0,a_1,a_2,\ldots,0)$ be a sequence of nonnegative integers which strictly decreases until $0$. Then there exists a graph that be d...

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

Problem What is the largest connected planar graph of minimum degree 3 which has everywhere positive combinatorial curvature, but is not a prism or an...

Conjecture Suppose $G$ is a finite group, and $n$ is a positive integer dividing $|G|$. Suppose that $G$ has exactly $n$ solutions to $x^{n} = 1$. Do...

Conjecture Is the theory of the real numbers with the exponential function decidable?...

Question How many steps does it take the sixth Goodstein sequence to terminate?...

Question Can either of the following be expressed in fixed-point logic plus counting: - Given a graph, does it have a perfect matching, i.e., a set $...

Question - Does ${<}\text{-invariant\:FO} = \text{FO}$ hold over graphs of bounded tree-width? - Is ${<}\text{-invariant\:FO}$ included in $\text{MSO...

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

Let $C$ be a class of finite relational structures. We denote by $f_C(n)$ the number of structures in $C$ over the labeled set $\{0, \dots, n-1 \}$. F...

Question Is the MSO-alternation hierarchy strict for pictures that are balanced, in the sense that the width and the length are polynomially (or linea...

Question Positive Horn logic (pH) is the fragment of FO involving exactly $\exists, \forall, \wedge, =$. Does the fragment $pH \wedge \neg pH$ have th...

Conjecture For every $\varepsilon > 0$ there is $\delta > 0$ such that, for every large $n$, there are $n$-vertex graphs $G$ and $H$ such that $G \equ...

Conjecture Every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product....

Conjecture Let $a > b > 0$ be integers. Set $b_1 = b$ and $b_{i+1} = {a \bmod {b_i}}$ for $i \geq 0$. Eventually we have $b_{n+1} = 0$; put $P(a,b) = ...

Definition A set of m positive integers $\{a_1, a_2, \dots, a_m\}$ is called a Diophantine $m$-tuple if $a_i\cdot a_j + 1$ is a perfect square for all...

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}}\equiv...