Cycle double coverIn graph-theoretic mathematics, a cycle double cover is a collection of cycles in an undirected graph that together include each edge of the graph exactly twice. For instance, for any polyhedral graph, the faces of a convex polyhedron that represents the graph provide a double cover of the graph: each edge belongs to exactly two faces. It is an unsolved problem, posed by George Szekeres and Paul Seymour and known as the cycle double cover conjecture, whether every bridgeless graph has a cycle double cover.
Lovász conjectureIn graph theory, the Lovász conjecture (1969) is a classical problem on Hamiltonian paths in graphs. It says: Every finite connected vertex-transitive graph contains a Hamiltonian path. Originally László Lovász stated the problem in the opposite way, but this version became standard. In 1996, László Babai published a conjecture sharply contradicting this conjecture, but both conjectures remain widely open. It is not even known if a single counterexample would necessarily lead to a series of counterexamples.
Extremal graph theoryExtremal graph theory is a branch of combinatorics, itself an area of mathematics, that lies at the intersection of extremal combinatorics and graph theory. In essence, extremal graph theory studies how global properties of a graph influence local substructure.
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.
Tietze's graphIn the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Möbius strip can be subdivided into six regions that all touch each other – three along the boundary of the strip and three along its center line – and therefore that graphs that are embedded onto the Möbius strip may require six colors.
Edge-transitive graphIn the mathematical field of graph theory, an edge-transitive graph is a graph G such that, given any two edges e_1 and e_2 of G, there is an automorphism of G that maps e_1 to e_2. In other words, a graph is edge-transitive if its automorphism group acts transitively on its edges. The number of connected simple edge-transitive graphs on n vertices is 1, 1, 2, 3, 4, 6, 5, 8, 9, 13, 7, 19, 10, 16, 25, 26, 12, 28 ... Edge-transitive graphs include all symmetric graph, such as the vertices and edges of the cube.
Flower snarkIn the mathematical field of graph theory, the flower snarks form an infinite family of snarks introduced by Rufus Isaacs in 1975. As snarks, the flower snarks are connected, bridgeless cubic graphs with chromatic index equal to 4. The flower snarks are non-planar and non-hamiltonian. The flower snarks J5 and J7 have book thickness 3 and queue number 2. The flower snark Jn can be constructed with the following process : Build n copies of the star graph on 4 vertices. Denote the central vertex of each star Ai and the outer vertices Bi, Ci and Di.
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.
Lovász numberIn graph theory, the Lovász number of a graph is a real number that is an upper bound on the Shannon capacity of the graph. It is also known as Lovász theta function and is commonly denoted by , using a script form of the Greek letter theta to contrast with the upright theta used for Shannon capacity. This quantity was first introduced by László Lovász in his 1979 paper On the Shannon Capacity of a Graph. Accurate numerical approximations to this number can be computed in polynomial time by semidefinite programming and the ellipsoid method.
Tensor product of graphsIn graph theory, the tensor 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 vertices (g,h) and math|(''g,h' ) are adjacent in G × H if and only if g is adjacent to g' in G, and h is adjacent to h' in H. The tensor product is also called the direct product, Kronecker product, categorical product, cardinal product, relational product, weak direct product, or conjunction'''.
