Eulerian cycles and paths are by far one of the most influential concepts in Graph Theory. However, what really are Eulerian cycles and paths, and how is the 18th-century path meaningful to the futuristic 21st century?
Eulerian Cycles and paths are by far one of the most influential concepts of graph theory in the world of mathematics and innovative technology. These circuits and paths were first discovered by Euler in 1736, therefore giving the name “Eulerian Cycles” and “Eulerian Paths.” When it comes to graph theory, understanding graphs and creating them are slightly more complex than it looks. There are many variables to consider, making them seem more like a puzzle than an actual problem. However, when we talk about Eulerian Cycles and Paths, it’s relatively easy to understand what’s going on.
An Eulerian Path is a path in a graph where each edge is visited exactly once. An Euler path can have any starting point with any ending point; however, the most common Euler paths lead back to the starting vertex. We can easily detect an Euler path in a graph if the graph itself meets two conditions: all vertices with non-zero degree edges are connected, and if zero or two vertices have odd degrees and all other vertices have even degrees. Do note that only one vertex with an odd degree is not possible in an undirected graph (Eulerian Paths are commonly found in undirected graphs) as the sum of all degrees is always even in an undirected graph. But you may ask, “how do we know if a vertex has an odd degree or an even degree?”. For those who don’t know about degrees in graphs, finding degrees of vertices is different than finding the degrees of typical angles. If the total number of edges of a vertex is odd, the vertex is said to have an odd degree. Though, if the total number of edges of a vertex is an even number, the vertex is said to have an even degree. This odd-even vertex condition allows us to understand if a given graph is Eulerian or not.
To know if a graph is Eulerian, or in other words, to know if a graph has an Eulerian cycle, we must understand that the vertices of the graph must be positioned where each edge is visited once and that the final edge leads back to the starting vertex. The Eulerian Cycle is essentially just an extended definition of the Eulerian Path. If it seems confusing, then picture it like this, “is it possible to draw the graph without lifting your pencil or pen (in…
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