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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Find the largest graphs with maximum degree d and diameter k. Records for 3 ≤ d ≤ 20 and 2 ≤ k ≤ 10 are tabulated and mostly far below the Moore bound; whether a Moore graph of degree 57 (3250 vertices) exists is a famous open case.
In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing.
Let G be a finite graph with minimum degree at least 3. Must G contain a cycle of length 2^k for some k ≥ 2?
C_27b is the highest possible chromatic number for any biplanar graph.
For 0 ≤ ρ ≤ 1, let C(ρ) denote the largest quantity such that any graph on n vertices and (ρ+o(1)) C(n, 2) edges will have at least (C(ρ)-o(1)) C(n, 3) triangles. What is C(ρ)?
Let C_7 denote the cycle graph on 7 vertices. We define C_9 to be the Shannon capacity of mathcal C_7: C_9 := Θ(mathcal C_7), where for a graph G, the Shannon capacity Θ(G) is defined by Θ(G) := sup_n ≥ 1 α(G^boxtimes n)^1/n.
A graphon is a symmetric measurable function W : [0,1]^2 → [0,1]. Given a graphon W and a finite graph H = (V(H),E(H)), the homomorphism density t(H,W) is defined as t(H,W) = ∫_[0,1]^V(H) Π_v,w ∈ E(H) W(x_v,x_w) Π_v ∈ V(H) dx_v.
Let G be a simple graph. The acyclic chromatic index χ_a'(G) of G is defined to be the least number of colors needed to color the edges of G so that no two edges coincident on the same vertex are homochromatic and there is no cycle whose edges are colored with only two colors.
