"an Eulerian trail (or Eulerian path) is a trail in a finite graph that visits every edge exactly once (allowing for revisiting vertices)."
Eulerian graphs have a circuit that visits every edge exactly once. They are often used in puzzles and games.
Basic Graph Theory: Understanding basic concepts, such as vertices, edges, and degrees, is crucial to understanding Eulerian graphs.
Eulerian Paths and Circuits: Eulerian paths and circuits are unique paths in a graph that include all edges. Understanding these concepts is essential to understanding Eulerian graphs.
The Euler Characteristic: The Euler characteristic is a measure of the topology of a graph. It is an important concept in the study of Eulerian graphs.
Graph Connectivity: Graph connectivity refers to the degree to which vertices in a graph are connected. Understanding graph connectivity is important when studying Eulerian graphs.
Hamiltonian Graphs: Hamiltonian graphs are graphs in which a Hamiltonian cycle can be found. These cycles are related to Eulerian circuits and are often studied alongside Eulerian graphs.
Digraphs: A digraph is a directed graph, in which the edges have a direction associated with them. Understanding digraphs and their properties is crucial to understanding Eulerian graphs.
Chinese Postman Problem: The Chinese Postman Problem involves finding the shortest path that a postman can take to deliver mail to all addresses on his route. This problem is similar to the Eulerian Path, and is often studied alongside Eulerian graphs.
Algorithmic Approaches to Finding Eulerian Paths and Circuits: There are several algorithmic approaches to finding Eulerian Paths and Circuits, including Fleury’s Algorithm, Hierholzer’s Algorithm and Kosaraju’s Algorithm. An understanding of these algorithms is important when studying Eulerian graphs.
Applications of Eulerian Graphs: Eulerian Graphs have many practical applications, including in the field of network analysis, transportation planning, and logistics. Understanding these applications is essential to understanding the real-world significance of Eulerian graphs.
Critique of Eulerian Graphs: There are critiques of the Eulerian graph, such as the fact that they assume every edge is traveled in the same amount of time. It is important to understand these critiques when studying Eulerian graphs.
Simple Eulerian Graph: A connected graph is said to be simple Eulerian if and only if every vertex of the graph has an even degree.
Multigraph Eulerian Graph: A connected graph is said to be multigraph Eulerian if and only if it contains more than one edge between two vertices.
Pseudo-Eulerian Graphs: These graphs have exactly two vertices with odd degree.
Semi-Eulerian Graphs: These graphs contain an Eulerian trail or Eulerian circuit but are not Eulerian.
Semi-Eulerian planar Graphs: The Euler Trail passes through edges and only two network nodes are odd-order vertices.
Directed Eulerian graphs: These graphs are valid when a unique path is available from one node to another, in which the edges are directed and hold the ability to reach all nodes.
Complete graph: All nodes are connected with each other in a complete graph as vertices, so this is always a Eulerian circuit.
Trivial Eulerian Graphs: In such graphs, there is only one vertex with an even degree.
Directed semi-Eulerian Graph: These are Directed Graphs that have at most one vertex of odd degree.
Vertex-Transitive eulerian Graph: These graphs are Eulerian and contain various permutations of vertices that lead to isomorphism.
Regular Eulerian Graph: A graph where every vertex has the same degree is called a regular graph.
Weak Eulerian Graphs: It is a graph using directed edges in which an Euler cycle exists when disregarding the edge direction.
"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 Bridges of Königsberg problem in 1736."
"A necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree."
"The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer."
"Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even degree."
"These definitions coincide for connected graphs."
"For the existence of Eulerian trails, it is necessary that zero or two vertices have an odd degree."
"The Königsberg graph is not Eulerian."
"A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian."
"They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736."
"Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree."
"The first complete proof of this latter claim was published posthumously in 1873 by Carl Hierholzer."
"Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even degree."
"These definitions coincide for connected graphs."
"For the existence of Eulerian trails, it is necessary that zero or two vertices have an odd degree."
"The Königsberg graph is not Eulerian."
"A graph that has an Eulerian trail but not an Eulerian circuit is called semi-Eulerian."
"They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736."
"Euler proved that a necessary condition for the existence of Eulerian circuits is that all vertices in the graph have an even degree."