Unsolved Problems

Question (Noel, 2013) Does there exist a function $f(k)=o(k^2)$ such that for every graph $G$, $$ \text{ch}\left(G^2\right)\leq f\left(\chi\left(G^2\r...

Conjecture If $G$ is a non-empty graph containing no induced odd cycle of length at least $5$, then there is a $2$-vertex colouring of $G$ in which no...

Question Given $a,b\geq2$, what is the smallest integer $t\geq0$ such that $\chi_\ell(K_{a,b}+K_t)= \chi(K_{a,b}+K_t)$?...

Conjecture Every $K_t$-minor-free graph is $c t$-list-colourable for some constant $c\geq1$....

Conjecture If $\chi(G)>k$, then $G$ contains at least $\frac{(k+1)(k-1)!}{2}$ cycles of length $0\bmod k$....

Conjecture If $G$ is an orientation of a simple planar graph, then there is a partition of $V(G)$ into $\{X_1,X_2\}$ so that the graph induced by $X_i...

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

An antidirected tree is an orientation of a tree in which every vertex has either indegree 0 or outdergree 0. Conjecture Let $D$ be a digraph. If $|A...

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...

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 transit...

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...

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 cyc...

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?...