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In graph theory, an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices). Similarly, an Eulerian circuit or Eulerian cycle is an Eulerian trail that starts and ends on the same vertex. They were first discussed by Leonhard Euler while solving the famous Seven ...
In graph theory, an n -dimensional De Bruijn graph of m symbols is a directed graph representing overlaps between sequences of symbols. It has mn vertices, consisting of all possible length-n sequences of the given symbols; the same symbol may appear multiple times in a sequence. For a set of m symbols S = {s1, …, sm}, the set of vertices is:
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, [1] laid the foundations of graph theory and prefigured the idea of topology.
In graph theory, the handshaking lemma is the statement that, in every finite undirected graph, the number of vertices that touch an odd number of edges is even. For example, if there is a party of people who shake hands, the number of people who shake an odd number of other people's hands is even. [1] The handshaking lemma is a consequence of ...
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a cycle that visits each vertex exactly once. A Hamiltonian path that starts and ends at adjacent vertices can be completed by adding ...
In graph theory, a branch of mathematics, a cycle basis of an undirected graph is a set of simple cycles that forms a basis of the cycle space of the graph. That is, it is a minimal set of cycles that allows every even-degree subgraph to be expressed as a symmetric difference of basis cycles. A fundamental cycle basis may be formed from any ...
BEST theorem. In graph theory, a part of discrete mathematics, the BEST theorem gives a product formula for the number of Eulerian circuits in directed (oriented) graphs. The name is an acronym of the names of people who discovered it: N. G. de Bruijn, Tatyana Ehrenfest, Cedric Smith and W. T. Tutte.
Path (graph theory) A three-dimensional hypercube graph showing a Hamiltonian path in red, and a longest induced path in bold black. In graph theory, a path in a graph is a finite or infinite sequence of edges which joins a sequence of vertices which, by most definitions, are all distinct (and since the vertices are distinct, so are the edges).