Unsolved Problems

For a graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the ch...

Conjecture For every positive integer $t$, every (loopless) graph $G$ with $\chi(G) \ge t$ immerses $K_t$....

The Odd Distance Graph, denoted ${\mathcal O}$, is the graph with vertex set ${\mathbb R}^2$ and two points adjacent if the distance between them is a...

Conjecture Let $G$ be a simple graph with $n$ vertices and list chromatic number $\chi_\ell(G)$. Suppose that $0\leq t\leq \chi_\ell$ and each vertex ...

Let $G$ be a simple graph, and for every list assignment $\mathcal{L}$ let $\lambda_{\mathcal{L}}$ be the maximum number of vertices of $G$ which are ...

Conjecture If $G,H$ are simple finite graphs, then $\chi(G \times H) = \min \{ \chi(G), \chi(H) \}$. Here $G \times H$ is the tensor product (also ca...

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 ...

Conjecture Every planar graph is acyclically 5-choosable....

Conjecture If $G$ is a simple graph which is the union of $k$ pairwise edge-disjoint complete graphs, each of which has $k$ vertices, then the chromat...

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

Conjecture Any oriented graph has a vertex whose outdegree is at most its second outdegree....

For any simple digraph $G$, we let $\gamma(G)$ be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and $\beta(G)$...

Conjecture Every digraph with chromatic number at least $2k-2$ contains every oriented tree of order $k$ as a subdigraph....

Conjecture Every oriented graph with minimum outdegree $k$ contains a directed path of length $2k$....

Conjecture Every digraph with at least one directed cycle has an arc whose reversal reduces the number of directed cycles....

Problem Is there a minimum integer $f(d)$ such that the vertices of any digraph with minimum outdegree $d$ can be partitioned into two classes so that...

Conjecture Every strong oriented graph in which each vertex has indegree and outdegree at least $d$ contains a directed cycle of length at least $2d+1...

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_{...

Conjecture For every $k$, there exists an integer $f(k)$ such that if $D$ is a digraph whose arcs are colored with $k$ colors, then $D$ has a $S$ set ...

In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same ...

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....

Conjecture For every fixed graph $H$, there exists a constant $\delta(H)$, so that every graph $G$ without an induced subgraph isomorphic to $H$ conta...

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....

Conjecture For every fixed $k\geq2$ and fixed colouring $\chi$ of $E(K_k)$ with $m$ colours, there exists $\varepsilon>0$ such that every colouring of...

Conjecture Let $H$ be an $r$-uniform $r$-partite hypergraph. If $\nu$ is the maximum number of pairwise disjoint edges in $H$, and $\tau$ is the size ...

Conjecture Every infinite graph is a proper minor of itself....

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?...

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 tai...

If $G$ is a graph, we say that a partition of $V(G)$ is unfriendly if every vertex has at least as many neighbors in the other classes as in its own. ...

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 matchi...

Call a graph an $(\aleph_0,\aleph_1)$-graph if it has a bipartition $(A,B)$ so that every vertex in $A$ has degree $\aleph_0$ and every vertex in $B$ ...

Conjecture If $G$ is a simple loopless triangulation of an orientable surface, then the dual of $G$ is 3-edge-colorable....

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 ...

The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane. Conjecture $\displaystyle cr(K_n) = \frac ...

The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane. Conjecture $\displaystyle cr(K_{m,n}) = \f...

We let $cr(G)$ denote the crossing number of a graph $G$. Conjecture Every graph $G$ with $\chi(G) \ge t$ satisfies $cr(G) \ge cr(K_t)$....

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...

We say that a set $P \subseteq {\mathbb R}^2$ is $n$-universal if every $n$ vertex planar graph can be drawn in the plane so that each vertex maps to ...

Conjecture Is there a graph $G$ that is a minor-minimal obstruction for two non-orientable surfaces?...

Problem Does there exist a finitely presented group of intermediate growth?...

Problem Find a constant $k$ such that for any $d$ there is a sequence of tautologies of depth $k$ that have polynomial (or quasi-polynomial) size proo...

Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for...

Problem Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set $$ m(A) = - \min_x \sum_{a \in A} \cos(ax). $$ What is $m(n) = \min_A...

Call a polynomial $p \in {\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational. Conjecture There doe...

Conjecture $\pi$ and $e$ are algebraically independent...

Question Is Euler-Mascheroni constant an transcendental number?...

Conjecture Are all Fermat Numbers $$ F_n = 2^{2^{n } } + 1 $$ Square-Free?...

Conjecture A Fermat prime is a Fermat number $$ F_n = 2^{2^n } + 1 $$ that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_...

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?...

Say that a set $S \subseteq {\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums. Conjecture There exists a fixed cons...