Unsolved Problems

Can every graph with maximum degree Δ be decomposed into at most ⌈(Δ+1)/2⌉ linear forests?...

Does every finite connected vertex-transitive graph contain a Hamiltonian path?...

For which 2-regular graphs H can the complete graph be decomposed into edge-disjoint copies of H?...

What is the maximum pathwidth of an n-vertex cubic graph?...

What is the longest induced path in an n-dimensional hypercube graph?...

Does every (2n-2)-vertex tournament contain every n-vertex oriented tree?...

Can the edges of any graph be covered by at most 2ν triangles, where ν is the maximum size of a triangle packing?...

Does every countable graph admit a partition where every vertex has at least as many neighbors outside its part as inside?...

What is the maximum number of edges in a bipartite graph on (m,n) vertices with no complete bipartite subgraph $K_{s,t}$?...

For the Cartesian product of graphs $G \square H$, is the domination number at least $\gamma(G) \cdot \gamma(H)$?...

Do complete k-uniform hypergraphs admit Hamiltonian decompositions into tight cycles?...

Are there graphs on n vertices requiring more than floor(n/2) copies of each letter for word-representation?...

Characterize which planar graphs are word-representable....

Characterize word-representable graphs in terms of forbidden induced subgraphs....

Characterize word-representable near-triangulations containing K₄....

Classify graphs with representation number exactly 3....

Among bipartite graphs, do crown graphs require the longest word-representants?...

Is the line graph of a non-word-representable graph always non-word-representable?...

Which hard graph problems can be efficiently solved by translating graphs to their word representations?...

If every edge has imbalance ≥1, is the multiset of edge imbalances always graphic?...

Do slowly-growing hereditary graph families admit implicit representations?...

For r-partite r-uniform hypergraphs, is the vertex cover number at most (r-1) times the matching number?...

Does every oriented graph have a vertex with at least as many vertices at distance 2 as at distance 1?...

Is the bondage number of a graph always ≤ 3Δ/2, where Δ is the maximum degree?...

Does every bridgeless graph have a nowhere-zero 5-flow?...

Does every Petersen-minor-free bridgeless graph have a nowhere-zero 4-flow?...

Is the minimum dicut size equal to the maximum number of disjoint dijoins in a directed graph?...

Relate the order of the center of the Steinberg group of the ring of integers to the Dedekind zeta function....

If a polynomial of degree d over a field of characteristic 0 shares a factor with each of its first d-1 derivatives, must it be $(x-a)^d$?...

Can every finite von Neumann algebra be embedded into an ultrapower of the hyperfinite II₁ factor?...

Is $\|f(A)\| \leq 2 \sup_{z \in W(A)} |f(z)|$ for any matrix A and analytic function f on the numerical range W(A)?...

Characterize the determinant of the sum of two normal matrices....

Does every group with cohomological dimension 2 have a 2-dimensional Eilenberg-MacLane space K(G,1)?...

Are the assembly maps in algebraic K-theory and L-theory isomorphisms?...

Is every finite lattice isomorphic to the congruence lattice of some finite algebra?...

Does a Hadamard matrix of order 4k exist for every positive integer k?...

If a ring has no nil two-sided ideal besides {0}, does it also have no nil one-sided ideal besides {0}?...

Does there exist a perfect cuboid—a rectangular parallelepiped with integer edges, face diagonals, and space diagonal?...

Given n bases of an n-dimensional matroid, can we find n disjoint rainbow bases?...

Is every simple group with a stable first-order theory an algebraic group over an algebraically closed field?...

Can all regular languages be expressed with generalized regular expressions having bounded star height?...

For which number fields is there an algorithm to determine if a Diophantine equation has solutions?...

Does every complete first-order theory in a countable language have countably many, $\aleph_0$, or $2^{\aleph_0}$ countable models?...

Is the theory of the real numbers with addition, multiplication, and exponentiation decidable?...

Is every infinite field with a stable first-order theory separably closed?...

Do Henson graphs have the finite model property?...

Does there exist an o-minimal first-order theory with a trans-exponential (rapid growth) function?...

Is every infinite minimal field of characteristic zero algebraically closed?...

Determine the structure of Keisler's order on first-order theories....

For simply connected semisimple algebraic groups over fields of cohomological dimension ≤2, is $H^1(F,G) = 0$?...