TreeToCertificate[t] returns a binary string that is a certificate for the tree t such that trees have the same certificate if and only if they are isomorphic.

TriangleInequalityQ[g] yields True if the weights assigned to the edges of graph g satisfy the triangle inequality.

Turan[n, p] constructs the Turán graph, the extremal graph on n vertices that does not contain CompleteGraph[p].

TutteGraph returns the Tutte graph, the first known example of a 3-connected, 3-regular, planar graph that is non-Hamiltonian.

TwoColoring[g] finds a two-coloring of graph g if g is bipartite. It returns a list of the labels 1 and 2 corresponding to the vertices.

Undirected is an option to inform certain functions that the graph is undirected.

UndirectedQ[g] yields True if graph g is undirected.

UnionSet[a, b, s] merges the sets containing a and b in union-find data structure s.

Uniquely3ColorableGraph returns a 12-vertex, triangle-free graph with chromatic number 3 that is uniquely 3-colorable.

UnitransitiveGraph returns a 20-vertex, 3-unitransitive graph, discovered by Coxeter, that is not isomorphic to a 4-cage or a 5-cage.