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Kneser–Poulsen conjecture ★★★
Keywords: pushing disks
Wide partition conjecture ★★
Keywords:
3-accessibility of Fibonacci numbers ★★
Keywords: Fibonacci numbers; monochromatic diffsequences
Simplexity of the n-cube ★★★
Author(s):
Keywords: cube; decomposition; simplex
Crossing sequences ★★
Author(s): Archdeacon; Bonnington; Siran
Then there exists a graph that be drawn on a surface with orientable (nonorientable, resp.) genus with crossings, but not with less crossings.
Keywords: crossing number; crossing sequence
The Crossing Number of the Complete Graph ★★★
Author(s):
The crossing number of is the minimum number of crossings in all drawings of in the plane.
Keywords: complete graph; crossing number
The Crossing Number of the Hypercube ★★
The crossing number of is the minimum number of crossings in all drawings of in the plane.
The -dimensional (hyper)cube is the graph whose vertices are all binary sequences of length , and two of the sequences are adjacent in if they differ in precisely one coordinate.
Keywords: crossing number; hypercube
Monochromatic reachability or rainbow triangles ★★★
Author(s): Sands; Sauer; Woodrow
In an edge-colored digraph, we say that a subgraph is rainbow if all its edges have distinct colors, and monochromatic if all its edges have the same color.
Keywords: digraph; edge-coloring; tournament
Rank vs. Genus ★★★
Author(s): Johnson
Keywords:
The Hodge Conjecture ★★★★
Author(s): Hodge
Keywords: Hodge Theory; Millenium Problems
2-accessibility of primes ★★
Keywords: monochromatic diffsequences; primes
Non-edges vs. feedback edge sets in digraphs ★★★
Author(s): Chudnovsky; Seymour; Sullivan
For any simple digraph , we let be the number of unordered pairs of nonadjacent vertices (i.e. the number of non-edges), and be the size of the smallest feedback edge set.
Keywords: acyclic; digraph; feedback edge set; triangle free
Tarski's exponential function problem ★★
Author(s): Tarski
Keywords: Decidability
Counting 3-colorings of the hex lattice ★★
Author(s): Thomassen
Keywords: coloring; Lieb's Ice Constant; tiling; torus
Dense rational distance sets in the plane ★★★
Author(s): Ulam
Keywords: integral distance; rational distance
Negative association in uniform forests ★★
Author(s): Pemantle
Keywords: forest; negative association
Wall-Sun-Sun primes and Fibonacci divisibility ★★
Author(s):
Equivalently:
Total Colouring Conjecture ★★★
Author(s): Behzad
Keywords: Total coloring
Edge Reconstruction Conjecture ★★★
Author(s): Harary
Every simple graph with at least 4 edges is reconstructible from it's edge deleted subgraphs
Keywords: reconstruction