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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.

7 votes

Lagrange four-squares theorem: efficient algorithm with units modulo a prime?

There is also an unconditional deterministic polynomial-time algorithm to find $x,y,z,w \in \mathbf{F}_p^\times$ such that $x^2+y^2+z^2+w^2=n$, given any $n \in \mathbf{Z}$ and any prime $p \ge 7$. F …
Bjorn Poonen's user avatar
  • 23.8k
14 votes

A graph on irrationals where p is adjacent to q if p^q or q^p is rational.

The answers are that $|E(G)|=2^{\aleph_0}$ and that every vertex of $G$ has degree $\aleph_0$. Proof: The positive real solutions to $x^y=2$ form a curve of cardinality $2^{\aleph_0}$, and at most …
Bjorn Poonen's user avatar
  • 23.8k
41 votes
Accepted

Bringing Number and Graph Theory Together: A Conjecture on Prime Numbers

Theorem: Schinzel's hypothesis H implies the conjecture. Proof: Choose distinct primes $q_S > 100|G|$ indexed by the 2-element subsets $S$ of $G$. For each $i \in G$, let $Q_i$ be the set of $q_S$ …
Bjorn Poonen's user avatar
  • 23.8k