Search results
Results From The WOW.Com Content Network
A cycle double cover of the Petersen graph, corresponding to its embedding on the projective plane as a hemi-dodecahedron. In 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 ...
Every bridgeless graph has a nowhere-zero 5-flow. The converse of the 4-flow Conjecture does not hold since the complete graph K 11 contains a Petersen graph and a 4-flow. For bridgeless cubic graphs with no Petersen minor, 4-flows exist by the snark theorem (Seymour, et al 1998, not yet published).
It has been proven that every bridgeless graph has cycle k-cover for any integer even integer k≥4. For k=2, it is the well-known cycle double cover conjecture is an open problem in graph theory. The cycle double cover conjecture states that in every bridgeless graph, there exists a set of cycles that together cover every edge of the graph twice.
Bridge (graph theory) A graph with 16 vertices and six bridges (highlighted in red) An undirected connected graph with no bridge edges. In graph theory, a bridge, isthmus, cut-edge, or cut arc is an edge of a graph whose deletion increases the graph's number of connected components. [1] Equivalently, an edge is a bridge if and only if it is not ...
In the mathematical field of graph theory, the Petersen graph is an undirected graph with 10 vertices and 15 edges. It is a small graph that serves as a useful example and counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic graph with ...
Since the Petersen graph is cubic and bridgeless, it meets the conditions of Petersen's theorem. In the mathematical discipline of graph theory, Petersen's theorem, named after Julius Petersen, is one of the earliest results in graph theory and can be stated as follows: Petersen's Theorem. Every cubic, bridgeless graph contains a perfect matching.
The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle ...
Cycle (graph theory) A graph with edges colored to illustrate a closed walk, H–A–B–A–H, in green; a circuit which is a closed walk in which all edges are distinct, B–D–E–F–D–C–B, in blue; and a cycle which is a closed walk in which all vertices are distinct, H–D–G–H, in red. In graph theory, a cycle in a graph is a non ...