BipartiteMatchingAndCover[g] takes a bipartite graph g and returns a matching with maximum weight along with the dual vertex cover. If the graph is not weighted, it is ...

BipartiteMatching[g] gives the list of edges associated with a maximum matching in bipartite graph g. If the graph is edge weighted, then the function returns a matching with ...

CubeConnectedCycle[d] returns the graph obtained by replacing each vertex in a d-dimensional hypercube by a cycle of length d. Cube-connected cycles share many properties ...

DeleteCycle[g, c] deletes a simple cycle c from graph g. c is specified as a sequence of vertices in which the first and last vertices are identical. g can be directed or ...

Dihedral is an argument to the Polya-theoretic functions ListNecklaces, NumberOfNecklaces, and NecklacePolynomial, which count or enumerate distinct necklaces. Dihedral ...

FindCycle[g] finds a list of vertices that define a cycle in graph g.

FranklinGraph returns a 12-vertex graph that represents a 6-chromatic map on the Klein bottle. It is the sole counterexample to Heawood's map-coloring conjecture.

Girth[g] gives the length of a shortest cycle in a simple graph g.

GraphicQ[s] yields True if the list of integers s is a graphic sequence, and thus represents a degree sequence of some graph.

HamiltonianQ[g] yields True if there exists a Hamiltonian cycle in graph g, or in other words, if there exists a cycle that visits each vertex exactly once.