Unsolved Problems

Conjecture Every simple connected graph on $n$ vertices can be decomposed into at most $\frac{1}{2}(n+1)$ paths....

Problem Does every $3$-connected cubic graph on $3k$ vertices admit a partition into $k$ paths of length $2$?...

Conjecture The minimum $k$-norm of a path partition on a directed graph $D$ is no more than the maximal size of an induced $k$-colorable subgraph....

Question Are there any (0,2)-graphs with chromatic number exactly three?...

Conjecture A total coloring of a graph $G = (V,E)$ is an assignment of colors to the vertices and the edges of $G$ such that every pair of adjacent ve...

Problem Do there exist 4-regular 4-chromatic graphs of arbitrarily high girth?...

Conjecture The union of a $1$-degenerate graph (a forest) and a $2$-degenerate graph is $5$-colourable....

Conjecture If $G$ is the total graph of a multigraph, then $\chi_\ell(G)=\chi(G)$....

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 thr...

Conjecture There exists a fixed constant $c$ (probably $c=1$ suffices) so that every graft with minimum $T$-cut size at least $k$ contains a $T$-join ...

Conjecture Every simple graph with maximum degree $\Delta$ has a proper $(\Delta+2)$-edge-colouring so that every cycle contains edges of at least thr...

Conjecture Let $H$ be a simple $d$-uniform hypergraph, and assume that every set of $d-1$ points is contained in at most $r$ edges. Then there exists ...

Conjecture Suppose that $G$ is a $\Delta$-edge-critical graph. Suppose that for each edge $e$ of $G$, there is a list $L(e)$ of $\Delta$ colors. Then ...

Problem Which Steiner triple systems are universal?...

Conjecture Let $G$ be a loopless multigraph. Then the edge chromatic number of $G$ equals the list edge chromatic number of $G$....

An $r$-graph is an $r$-regular graph $G$ with the property that $|\delta(X)| \ge r$ for every $X \subseteq V(G)$ with odd size. Conjecture $\chi'(G) ...

The overfull parameter is defined as follows: $$ w(G) = \max_{H \subseteq G} \left\lceil \frac{ |E(H)| }{ \lfloor \tfrac{1}{2} |V(H)| \rfloor} \right\...

A strong edge-colouring of a graph $G$ is a edge-colouring in which every colour class is an induced matching; that is, any two vertices belonging to ...

Conjecture Let $M$ be an abelian group. Is the core of a Cayley graph (on some power of $M$ ) a Cayley graph (on some power of $M$ )?...

Question Let $G$ be a 3-regular graph that contains no cycle of length shorter than $g$. Is it true that for large enough~ $g$ there is a homomorphism...

Conjecture Every planar graph of girth $\ge 4k$ has a homomorphism to $C_{2k+1}$....

Conjecture If $G$ is a cubic graph not containing a triangle, then it is possible to color the edges of $G$ by five colors, so that the complement of ...

Question Is there an algorithm that decides, for input graphs $G$ and $H$, whether there exists a homomorphism from $G$ to $H$ in time $O(c^{|V(G)|+|...

Let $G = (V, E)$ be a graph. If $p$ and $q$ are two integers, a $(p,q)$-colouring of $G$ is a function $c$ from $V$ to $\{0,\dots,p-1\}$ such that $q ...

Conjecture All trees are graceful...

Question What is the maximum edge density of a graph which has a good edge labeling? We say that a graph is good-edge-labeling critical, if it has no...

Conjecture Every bridgeless graph has a nowhere-zero 5-flow....

Conjecture Every bridgeless graph with no Petersen minor has a nowhere-zero 4-flow....

Conjecture Every 4-edge-connected graph has a nowhere-zero 3-flow....

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

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

Conjecture For every graph $G$ with no bridge, there exist three disjoint sets $A_1,A_2,A_3 \subseteq E(G)$ with $A_1 \cup A_2 \cup A_3 = E(G)$ so tha...

Conjecture Let $M,M'$ be abelian groups and let $B \subseteq M$ and $B' \subseteq M'$ satisfy $B=-B$ and $B' = -B'$. If there is a homomorphism from $...

Conjecture All real roots of nonzero flow polynomials are at most 4....

Conjecture For every graph $G$ without a bridge, there is a flow $\phi: E(G) \rightarrow S^2 = \{ x \in {\mathbb R}^3: |x| = 1 \}$. Conjecture There ...

If $G$, $H$ are graphs, a function $f: E(G) \rightarrow E(H)$ is called cycle-continuous if the pre-image of every element of the (binary) cycle space...

A nowhere-zero $r$-flow $(D(G),\phi)$ on $G$ is an orientation $D$ of $G$ together with a function $\phi$ from the edge set of $G$ into the real numbe...

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 $r$ th...

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...

Problem Does every triangle-free planar graph of maximum degree three have circular chromatic number at most $\frac{20}{7}$?...

An oriented colouring of an oriented graph is assignment $c$ of colours to the vertices such that no two arcs receive ordered pairs of colours $(c_1,c...

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...

Problem Find $\lim_{n \rightarrow \infty} (\chi( H_n, 3)) ^{ 1 / |V(H_n)| }$....

Let ${\mathcal O}$ denote the graph with vertex set consisting of all lines through the origin in ${\mathbb R}^3$ and two vertices adjacent in ${\math...

A connected simple graph $G$ is called double-critical, if removing any pair of adjacent vertexes lowers the chromatic number by two. Conjecture $K_n...

Conjecture A triangle-free graph with maximum degree $\Delta$ has chromatic number at most $\ceil{\frac{\Delta}{2}}+2$....