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The Petersen graph is a well known non-Hamiltonian graph, but all odd graphs for are known to have a Hamiltonian cycle. [17] As the odd graphs are vertex-transitive , they are thus one of the special cases with a known positive answer to Lovász' conjecture on Hamiltonian cycles in vertex-transitive graphs.
See Families of sets for related families of non-graph combinatorial objects, graphs for individual graphs and graph families parametrized by a small number of numeric parameters, and graph theory for more general information about graph theory. See also Category:Graph operations for graphs distinguished for the specific way of their construction
The Petersen graph is the smallest snark. The flower snark J 5 is one of six snarks on 20 vertices.. In the mathematical field of graph theory, a snark is an undirected graph with exactly three edges per vertex whose edges cannot be colored with only three colors.
Parametric Graphs: Yes; Implicit Polynomials: Yes; Web Export: all constructions exportable as web pages as a Java applet; Macros: usable both as tools with the mouse and as commands in the input field; Animation: Yes; Spreadsheet: Yes, the cells can contain any GeoGebra object (numbers, points, functions etc.) Dynamic text: Yes (including LaTeX)
Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
It was proved for specific types of graphs, such as wheels, [8] complete graphs, [9] complete bipartite graphs, and graphs with a local symmetry. [10] It was also proved in the limit p → 1 for any graph. [11] [12] Counterexamples for generalizations of the bunkbed conjecture have been published for site percolation, hypergraphs, and directed ...
Graph pebbling is a mathematical game played on a graph with zero or more pebbles on each of its vertices. 'Game play' is composed of a series of pebbling moves. A pebbling move on a graph consists of choosing a vertex with at least two pebbles, removing two pebbles from it, and adding one to an adjacent vertex (the second removed pebble is discarded from play). π(G), the pebbling number of a ...
In graph theory, a branch of mathematics, the cop number or copnumber of an undirected graph is the minimum number of cops that suffices to ensure a win (i.e., a capture of the robber) in a certain pursuit–evasion game on the graph.