TxGraffiti Conjecture 3: independent domination versus saturation (regular)
TxGraffiti Conjecture 3: for every r-regular graph G (r ≥ 1), i(G) ≤ μ^(G). This conjecture is open**.
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Open problems
Each problem states how progress is verified and what counts as a contribution. Besides the problems curated here, the catalogue includes open conjectures from Formal Conjectures (with Lean statements), optimization constants and the AlphaEvolve problems. Know one that belongs here? Propose a problem.
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Undirected circular permutations with adjacent products minus one all prime
Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.
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Unique Crystal Components
If n = ab is a crystal, then there are no other pairs of positive integers c, d > 1, different from the couple a, b, such that n = cd and B(c, d) ∈ ℕ, i.e., the components of the crystals are unique.
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Value of n-th cyclotomic polynomial at n
a(28341) is divisible by 283411^2. What is the next n such that a(n) is not squarefree?
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Values m = σ(k) (0 < m < 2n) with 2n - 1 - m and 2n - 1 + m prime
Conjecture (i): a(n) > 0 for all n > 3.
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Van Eck's sequence (OEIS A181391)
OEIS A181391 conjectures: "every number eventually appears."
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Vaught conjecture
The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite, aleph_0 or 2^aleph_0.
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VCₙ dimension of convex sets in ℝⁿ, ℝⁿ⁺¹, ℝⁿ⁺²
Every convex set in ℝ^3 has VC_2 dimension at most 2.
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Vizing's conjecture (1968)
Vizing's conjecture (1968). For all finite simple graphs G and H, the domination number of the Cartesian (box) product satisfies γ(G square H) ≥ γ(G) γ(H).
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Ways of writing n as an ordered sum of a triangular, square, and pentagonal number
It is conjectured that a(n) is always positive (Conjecture 1.1 of Sun (2009)).
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Ways to write n = k + m with p_k^2 - 2, p_m^2 - 2, and p_p_m^2 - 2 all prime
Conjecture: a(n) > 0 for all n > 1.
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Ways to write n = x + y + z with six linear and quadratic forms prime
Conjecture: a(n) > 0 for all n > 5.
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Ways to write n = x + y with 2^x + y(y+1)/2 prime
Conjecture: a(n) > 0 for all n > 1.
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Ways to write n = x + y with nx + y and ny - x both prime
Conjecture (i), part 1: a(n) > 0 for all n > 2.
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Ways to write n as the sum of three elements of ⌊ T(x)/3 ⌋ : x ≥ 1
Conjecture: For any positive integer m, every nonnegative integer n can be written as ⌊ T(x)/m ⌋ + ⌊ T(y)/m ⌋ + ⌊ T(z)/m ⌋ with x, y, z nonnegative integers.
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Weak tiling problems
Problem 4.2. Let Ω ⊂ ℝ be a finite union of three or more intervals. If Ω weakly tiles its complement, must it also tile its complement properly?
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Wieferich primes
There are infinitely many Wieferich primes.
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Wilson primes
There are infinitely many Wilson primes.
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Wolstenholme numbers
Conjecture: for n > 3, gcd(n, a(n-1)) = A089026(n). - Amiram Eldar and Thomas Ordowski, Jul 28 2019
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Wolstenholme Prime
It is conjectured that there are infinitely many Wolstenholme primes. Reference: Wikipedia
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Woodall Primes
There are infinitely many prime numbers of the form k * 2 ^ k - 1 for k > 1.
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Written on the Wall II - Conjecture 100
WOWII Conjecture 100 (status O): For a simple connected graph G, α(G) ≤ ⌈(max_v l(v) + 0.5 · degreeL2Norm(Gᶜ)) / 2⌉ where α(G) = G.indepNum is the independence number, max_v l(v) is the maximum over all vertices of the independence number of the neighbourhood (in G), and degreeL2Norm(Gᶜ) is the…
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Written on the Wall II - Conjecture 133
WOWII Conjecture 133: For a simple connected graph G, path(G) ≥ rad(G) + (avg_v l(v))^cC_4(G), where path(G) is the path number of the graph (number of vertices of a largest induced path), rad(G) is the radius (minimum eccentricity, as a natural number), avg_v l(v) = l(G) is the average…
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Written on the Wall II - Conjecture 189
WOWII Conjecture 189 (status O): For a simple connected graph G, if max_v distEven(v) ≤ 1 + σ(G), then G has a Hamiltonian path. Here distEven(v) is the number of vertices at even distance from v, and σ(G) is the second-smallest degree of G's degree sequence (per WOWII defEntry 65).
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Written on the Wall II - Conjecture 19
WOWII Conjecture 19 If G is connected then the size b(G) of a largest induced bipartite subgraph satisfies b(G) ≥ FLOOR((∑ ecc(v))/(|V|) + sSup (range (l G))), where ecc(v) denotes eccentricity and l(G) is the independence number of neighbourhoods.
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Written on the Wall II - Conjecture 198a
WOWII Conjecture 198a For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path. Here b(G) is the number of vertices in a largest induced bipartite subgraph, and ecc_avg(G) is the average eccentricity of G.
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Written on the Wall II - Conjecture 199
WOWII Conjecture 199 For a simple connected graph G, if tree(G) - 2 ≤ κ(G), then G has a Hamiltonian path. The nontrivial vertex type encodes the source assumption n > 1. Here tree(G) is the number of vertices in a largest induced tree and κ(G) = vertexConnectivity G is the vertex connectivity of G.
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Written on the Wall II - Conjecture 40
WOWII Conjecture 40 For a nontrivial connected graph G the size f(G) of a largest induced forest satisfies f(G) ≥ ceil((p(G) + b(G) + 1)/2) where p(G) is the path cover number and b(G) is the largest induced bipartite subgraph size.
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Written on the Wall II - Conjecture 61
WOWII Conjecture 61 For a simple connected graph G, the size f(G) of a largest induced forest satisfies f(G) ≥ residue(G) + ⌈ diam(G) / 3 ⌉, where residue(G) is the Havel-Hakimi residue and diam(G) is the diameter of G.
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Written on the Wall II - Conjecture 72
WOWII Conjecture 72 For a simple connected graph G, tree(G) ≥ ⌈ecc_avg(G) + max_v λ(v)/3⌉ where tree(G) is the number of vertices in a largest induced tree, ecc_avg(G) is the average eccentricity of G, and max_v l(v) = max_v indepNeighborsCard G v is the maximum over all vertices of the…
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Written on the Wall II - Conjecture 84
WOWII Conjecture 84 (status O): For a simple connected graph G with positive minimum degree, tree(G) ≥ 2 · rad(G) / δ(G), where tree(G) is the number of vertices in a largest induced subtree, rad(G) is the radius, and δ(G) = G.minDegree is the minimum degree.
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Written on the Wall II - Conjecture 96
WOWII Conjecture 96 For a simple connected graph G on at least two vertices, α(G) ≤ 1 + min_v distEven(v) · (max_v l(v) - 1) where α(G) = G.indepNum is the independence number, distEven(v) is the number of vertices at even distance from v, and max_v l(v) is the maximum independence number of a…
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Zagier's Conjecture on Multiple Zeta Values
Zagier's conjecture The ℚ-dimension of the vector space spanned by all multiple zeta values of weight n equals d_n, where d_n is the Zagier dimension sequence satisfying d_0 = 1, d_1 = 0, d_2 = 1, and d_n = d_n-2 + d_n-3 for n ≥ 3.
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Zariski Cancellation
The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic 0 is cancellative.
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