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.

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.

UnrankBinarySubset[n, l] gives the n\[Null]^th subset of list l, listed in increasing order of integers corresponding to the binary representations of the subsets.

UnrankGrayCodeSubset[n, l] gives the n\[Null]^th subset of list l, listed in Gray code order.

UnrankSetPartition[r, s, k] finds a k-block set partition of s with rank r. UnrankSetPartition[r, n, k] finds a k-block set partition of {1, 2, ..., n} with rank r.

UnrankKSubset[m, k, l] gives the m\[Null]^th k-subset of set l, listed in lexicographic order.

UnrankPermutation[r, l] gives the r\[Null]^th permutation in the lexicographic list of permutations of list l. UnrankPermutation[r, n] gives the r\[Null]^th permutation in ...

UnrankRGF[r, n] returns a restricted growth function defined on the first n natural numbers whose rank is r.