WebThe Petersen graph has girth 5, diameter 2, edge chromatic number 4, chromatic number 3, and chromatic polynomial. The Petersen graph is a cubic symmetric graph and is nonplanar. The following elegant proof … WebA graph @C is symmetric if its automorphism group acts transitively on the arcs of @C, and s-regular if its automorphism group acts regularly on the set of s-arcs of @C. Tutte [W.T. Tutte, A family of cubical graphs, Proc. Cambridge Philos. Soc. 43 (...
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WebThe Hoffman-Singleton theorem states that any Moore graph with girth 5 must have degree 2, 3, 7 or 57. The first three respectively are the pentagon, the Petersen graph, and the Hoffman-Singleton graph. The existence of a Moore graph with girth 5 and degree 57 is still open. A Moore graph is a graph with diameter \(d\) and girth \(2d + 1 ... The Petersen graph is strongly regular (with signature srg(10,3,0,1)). It is also symmetric, meaning that it is edge transitive and vertex transitive. More strongly, it is 3-arc-transitive: every directed three-edge path in the Petersen graph can be transformed into every other such path by a symmetry of the … See more 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 … See more The Petersen graph is nonplanar. Any nonplanar graph has as minors either the complete graph $${\displaystyle K_{5}}$$, or the See more The Petersen graph has chromatic number 3, meaning that its vertices can be colored with three colors — but not with two — such that no edge connects vertices of the same color. It has a list coloring with 3 colors, by Brooks' theorem for list colorings. See more The Petersen graph is the complement of the line graph of $${\displaystyle K_{5}}$$. It is also the Kneser graph $${\displaystyle KG_{5,2}}$$; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an … See more The Petersen graph has a Hamiltonian path but no Hamiltonian cycle. It is the smallest bridgeless cubic graph with no Hamiltonian cycle. It is See more The Petersen graph: • is 3-connected and hence 3-edge-connected and bridgeless. See the glossary See more • Exoo, Geoffrey; Harary, Frank; Kabell, Jerald (1981), "The crossing numbers of some generalized Petersen graphs", Mathematica Scandinavica, 48: 184–188, doi:10.7146/math.scand.a-11910. • Lovász, László (1993), Combinatorial Problems and Exercises (2nd … See more
WebMar 3, 2024 · Add a comment. 0. In this one page file is presented a simple algorithm (and even its pseudocode) based on BFS which computed the girth of a (connected undirected) graph G = ( V, E) in O ( V E) time. More fast algoritms for special graphs (in particular, sparse and planar) are discussed in this short CSTheory.SE thread. WebWe presents results and conjectures on the maximum number of cycles in cubic multigraphs of girth 2, 3, 4, respectively. For cubic cyclically 5-edge-connected graphs we have no conjecture but, we believe that the generalized Petersen graphs P(n, k) are relevant. We enumerate the hamiltonian and almost hamiltonian cycles in each P(n, 2).
WebThe girth is the length of the shortest cycle. This is probably the easier part of the question. What possible lengths can cycles have? If you just work your way up from the smallest possible length, you should be able to see which actually occur. WebSep 6, 2013 · The Petersen graph has diameter 2 and girth 5. In other words, the shortest cycle has length 5, and any two vertices are either adjacent or share a common vertex. …
WebMar 24, 2024 · The chromatic number of a graph G is the smallest number of colors needed to color the vertices of G so that no two adjacent vertices share the same color (Skiena 1990, p. 210), i.e., the smallest value of k …
WebMar 24, 2024 · The girth of a graphs is the length of one of its (if any) shortest graph cycles. Acyclic graphs are considered to have infinite girth (Skiena 1990, p. 191). The … library reshape in rWebFeb 20, 2024 · Video. The following graph G is called a Petersen graph and its vertices have been numbered from 0 to 9. Some letters have also been assigned to vertices of G, as can be seen from the following … library republic moA cubic graph (all vertices have degree three) of girth g that is as small as possible is known as a g-cage (or as a (3,g)-cage). The Petersen graph is the unique 5-cage (it is the smallest cubic graph of girth 5), the Heawood graph is the unique 6-cage, the McGee graph is the unique 7-cage and the Tutte eight cage is the unique 8-cage. There may exist multiple cages for a given girth. For instance there are three nonisomorphic 10-cages, each with 70 vertices: the Balaban 10-cage, the Harries … library resistor smd eagleWebMar 24, 2024 · A Moore graph of type is a regular graph of vertex degree and girth that contains the maximum possible number of nodes, namely. (Bannai and Ito 1973; Royle). Equivalently, it is a - cage graph, where is … library reserves ndWebThe dodecahedral graph is not Hamilton-connected and is the only known example of a vertex-transitive Hamiltonian graph (other than cycle graphs) that is not H-*-connected (Stan Wagon, pers. comm., May 20, 2013). The dodecahedral graph has 20 nodes, 30 edges, vertex connectivity 3, edge connectivity 3, graph diameter 5, graph radius 5, and … library reserve room clemsonlibrary reserve ucscWebJan 1, 2024 · It turns out that generalized Petersen graphs, though not generally pancyclic, miss only very few possible length of cycles. For k ∈ {2, 3}, we completely determine all possible cycle lengths in ... mci west dts training