The world's unsolved problems
993 open questions across 14 fields, gathered from Wikipedia. Pick one and start thinking.
Showing all 993 problems · surprise me
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Suppose Alice has a winning strategy for the vertex coloring game on a graph G with k colors. Does she have one for k+1 colors?
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The 1-factorization conjecture that if n is odd or even and k \geq n, n
1, respectively, then a k-regular graph with 2n vertices is 1-factorable.
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Cereceda's conjecture on the diameter of the space of colorings of degenerate graphs
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The Earth–Moon problem
what is the maximum chromatic number of biplanar graphs?
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The Erdős–Faber–Lovász conjecture on coloring unions of cliques
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The graceful tree conjecture that every tree admits a graceful labeling
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The Gyárfás–Sumner conjecture on χ-boundedness of graphs with a forbidden induced tree
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The Hadwiger conjecture relating coloring to clique minors
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The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
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Jaeger's Petersen-coloring conjecture
every bridgeless cubic graph has a cycle-continuous mapping to the Petersen graph
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The list coloring conjecture
for every graph, the list chromatic index equals the chromatic index
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The overfull conjecture that a graph with maximum degree \Delta(G) \geq n/3 is class 2 if and only if it has an overfull subgraph S…
The overfull conjecture that a graph with maximum degree \Delta(G) \geq n/3 is class 2 if and only if it has an overfull subgraph S satisfying \Delta(S) = \Delta(G).
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The total coloring conjecture of Behzad and Vizing that the total chromatic number is at most two plus the maximum degree
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The Albertson conjecture
the crossing number can be lower-bounded by the crossing number of a complete graph with the same chromatic number
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Conway's thrackle conjecture that thrackles cannot have more edges than vertices
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The GNRS conjecture on whether minor-closed graph families have \ell_1 embeddings with bounded distortion
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Harborth's conjecture
every planar graph can be drawn with integer edge lengths
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Negami's conjecture on projective-plane embeddings of graphs with planar covers
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The strong Papadimitriou–Ratajczak conjecture
every polyhedral graph has a convex greedy embedding
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Turán's brick factory problem
Is there a drawing of any complete bipartite graph with fewer crossings than the number given by Zarankiewicz?
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Guy's conjecture on the crossing number for complete graphs
Is there a drawing of any complete graph with fewer crossings than the number given by his upper bound?
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Universal point sets of subquadratic size for planar graphs
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Does there exist a conference graph for every number of vertices v>1 where v \equiv 1 \bmod 4 and v is an odd sum of two squares?
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Conway's 99-graph problem
does there exist a strongly regular graph with parameters (99,14,1,2)?
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Degree diameter problem
given two positive integers d, k, what is the largest graph of diameter k such that all vertices have degrees at most d?
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Jørgensen's conjecture that every 6-vertex-connected K_6-minor-free graph is an apex graph
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Does a Moore graph with girth 5 and degree 57 exist?
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Do there exist infinitely many strongly regular geodetic graphs, or any strongly regular geodetic graphs that are not Moore graphs?
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Barnette's conjecture
every cubic bipartite three-connected planar graph has a Hamiltonian cycle
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Gilbert–Pollak conjecture on the Steiner ratio of the Euclidean plane that the Steiner ratio is \sqrt{3}/2
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Data last refreshed 2026-10-20 03:01 UTC.