The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
488 results in Mathematics · clear · surprise me
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The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed…
The Oberwolfach problem on which 2-regular graphs have the property that a complete graph on the same number of vertices can be decomposed into edge-disjoint copies of the given graph.
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What is the largest possible pathwidth of an n-vertex cubic graph?
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The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted…
The reconstruction conjecture and new digraph reconstruction conjecture on whether a graph is uniquely determined by its vertex-deleted subgraphs.
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The snake-in-the-box problem
what is the longest possible induced path in an n-dimensional hypercube graph?
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Sumner's conjecture
does every (2n-2)-vertex tournament contain as a subgraph every n-vertex oriented tree?
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Szymanski's conjecture
every permutation on the n-dimensional doubly-directed hypercube graph can be routed with edge-disjoint paths.
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Tuza's conjecture
if the maximum number of disjoint triangles is \nu, can all triangles be hit by a set of at most 2\nu edges?
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The unfriendly partition conjecture
Does every countable graph have an unfriendly partition into two parts?
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Vizing's conjecture on the domination number of cartesian products of graphs
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Walescki's theorem for hypergraphs
Do complete k-uniform hypergraphs admit Hamiltonian decompositions into tight cycles?
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Zarankiewicz problem
how many edges can there be in a bipartite graph on a given number of vertices with no complete bipartite subgraphs of a given size?
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Are there any graphs on n vertices whose representation requires more than floor(n/2) copies of each letter?
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Characterise (non-)word-representable planar graphs
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Characterise word-representable graphs in terms of (induced) forbidden subgraphs.
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Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar…
Characterise word-representable near-triangulations containing the complete graph K4 (such a characterisation is known for K4-free planar graphs)
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Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be…
Classify graphs with representation number 3, that is, graphs that can be represented using 3 copies of each letter, but cannot be represented using 2 copies of each letter
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Is it true that out of all bipartite graphs, crown graphs require longest word-representants?
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Is the line graph of a non-word-representable graph always non-word-representable?
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Which (hard) problems on graphs can be translated to words representing them and solved on words (efficiently)?
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Delta-conjecture (1978)
consider a complete graph (V; E_1, \dots, E_d) each edge of which is colored by one of d colors such that there exists a triangle \Delta colored in three pairwise distinct colors. Then, in each chromatic component (V, E_i), i = 1, \dots, d, one can choose...
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The imbalance conjecture
If the imbalance for each edge of a graph is at least 1, is the multiset of all edge imbalances always graphic?
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The implicit graph conjecture on the existence of implicit representations for slowly-growing hereditary families of graphs
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Ryser's conjecture relating the maximum matching size and minimum transversal size in hypergraphs
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The second neighborhood problem
does every oriented graph contain a vertex for which there are at least as many other vertices at distance two as at distance one?
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Sidorenko's conjecture on homomorphism densities of graphs in graphons
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Teschner's bondage number conjecture
is the bondage number of a graph always less than or equal to 3/2 times its maximum degree?
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Tutte's conjectures
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Woodall's conjecture that the minimum number of edges in a dicut of a directed graph is equal to the maximum number of disjoint dijoins.
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The Cherlin–Zilber conjecture
A simple group whose first-order theory is stable in \aleph_0 is a simple algebraic group over an algebraically closed field.
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Generalized star height problem
can all regular languages be expressed using generalized regular expressions with limited nesting depths of Kleene stars?
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Data last refreshed 2026-10-20 03:01 UTC.