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A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree. These definitions coincide for connected graphs. [2]
In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg(v). The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula
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. [2]
1 Examples and types of graphs. 2 Graph coloring. ... Path (graph theory) Seven Bridges of Königsberg. Eulerian path; Three-cottage problem;
A spanning subgraph of a given graph G has the same set of vertices as G itself but, possibly, fewer edges. A graph G, or one of its subgraphs, is said to be Eulerian if each of its vertices has even degree (its number of incident edges). Every simple cycle in a graph is an Eulerian subgraph, but there may be others.
In mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices (also called nodes or points ) which are connected by edges (also called arcs , links or lines ).
When the graph has an Eulerian circuit (a closed walk that covers every edge once), that circuit is an optimal solution. Otherwise, the optimization problem is to find the smallest number of graph edges to duplicate (or the subset of edges with the minimum possible total weight) so that the resulting multigraph does have an Eulerian circuit. [1]
By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example's Boolean equation i.e. (x'y'z' + x'y'z) + (x'yz' + xyz') to just two terms: x'y' + yz'. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this ...