GraphIntersection[g_1, g_2, ...] constructs the graph defined by the edges that are in all the graphs g_1, g_2, ....

GraphProduct[g_1, g_2, ...] constructs the product of graphs g_1, g_2, and so forth.

GraphSum[g_1, g_2, ...] constructs the graph resulting from joining the edge lists of graphs g_1, g_2, and so forth.

GrayCodeKSubsets[l, k] generates k-subsets of l in Gray code order.

GrayGraph returns a 3-regular, 54-vertex graph that is edge-transitive but not vertex-transitive\[LongDash]the smallest known such example.

GroetzschGraph returns the smallest triangle-free graph with chromatic number 4. This is identical to MycielskiGraph[4].

HamiltonianPath[g] finds a Hamiltonian path in graph g if one exists. HamiltonianPath[g, All] gives all Hamiltonian paths of graph g.

HasseDiagram[g] constructs a Hasse diagram of the relation defined by directed acyclic graph g.

HeawoodGraph returns a smallest (6, 3)-cage, a 3-regular graph with girth 6.

HighlightedEdgeColors is an option to Highlight that determines which colors are used for the highlighted edges.