Crown graphIn graph theory, a branch of mathematics, a crown graph on 2n vertices is an undirected graph with two sets of vertices {u_1, u_2, ..., u_n} and {v_1, v_2, ..., v_n} and with an edge from u_i to v_j whenever i ≠ j. The crown graph can be viewed as a complete bipartite graph from which the edges of a perfect matching have been removed, as the bipartite double cover of a complete graph, as the tensor product K_n × K_2, as the complement of the Cartesian direct product of K_n and K_2, or as a bipartite Kneser graph H_n,1 representing the 1-item and (n – 1)-item subsets of an n-item set, with an edge between two subsets whenever one is contained in the other.
Cartesian product of graphsIn graph theory, the Cartesian product G □ H of graphs G and H is a graph such that: the vertex set of G □ H is the Cartesian product V(G) × V(H); and two vertices (u,u' ) and (v,v' ) are adjacent in G □ H if and only if either u = v and u' is adjacent to v' in H, or u' = v' and u is adjacent to v in G. The Cartesian product of graphs is sometimes called the box product of graphs [Harary 1969]. The operation is associative, as the graphs (F □ G) □ H and F □ (G □ H) are naturally isomorphic.
Graph homomorphismIn the mathematical field of graph theory, a graph homomorphism is a mapping between two graphs that respects their structure. More concretely, it is a function between the vertex sets of two graphs that maps adjacent vertices to adjacent vertices. Homomorphisms generalize various notions of graph colorings and allow the expression of an important class of constraint satisfaction problems, such as certain scheduling or frequency assignment problems.
Bipartite double coverIn graph theory, the bipartite double cover of an undirected graph G is a bipartite, covering graph of G, with twice as many vertices as G. It can be constructed as the tensor product of graphs, G × K_2. It is also called the Kronecker double cover, canonical double cover or simply the bipartite double of G. It should not be confused with a cycle double cover of a graph, a family of cycles that includes each edge twice. The bipartite double cover of G has two vertices u_i and w_i for each vertex v_i of G.
