Questions tagged [eulerian-path]
This tag is for questions relating to Eulerian paths in graphs. An "Eulerian path" or "Eulerian trail" in a graph is a walk that uses each edge of the graph exactly once. An Eulerian path is "closed" if it starts and ends at the same vertex.
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Worst Case Solution for directed Chinese Postman Problem
The Problem
Let $G$ be a directed graph with $n$ vertices. How long is a shortest circuit that visits every edge in the worst case? That is how long is the solution to the directed Chinese Postman ...
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Constructing Paths in a Connected Graph with Odd-Degree Vertices
I am working on a problem and would appreciate some insights or suggestions on how to approach it.
Problem:
Let G = (V, E) be a connected graph where n is the number of vertices in V that are of odd ...
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Spanning Trees with 1/2 the edges in a Eulerian Graph
I was attempting the following problem:
Let $G$ be a connected simple graph. (a) If $G$ is eulerian with an even number of vertices, then it has a spanning subgraph $G'$ such that every node $i$ has ...
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Finding an Eulerian path on complete graphs
I want to prove the following algorithm to find an Eulerian path in a complete graph:
In the complete graph $K_{n}$ where $n = 2k + 1, k \in \mathbb{Z}^+$, let us label the vertices $1, 2, ..., n$. ...
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counting Eulerian circuits on complete directed graph
I have a complete directed graph $G$ (including self-loops). How can I count the number of Eulerian circuits on $G$?
For example, in the simple case of $n=2$, there are clearly 4 Eulerian circuits. ...
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What are necessary and sufficient conditions to have a negative cycle in a directed graph with some negative edges? [closed]
Trying to test Johnson’s algorithm with over 100 vertices but it doesn’t work if there is a negative cycle. So I’m trying to write code to construct graphs with some negative weights (about 10% of the ...
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Lemma for proving the Euler's theorem
I was studying graph theory when a question came to my mind.
I am referring to a particular way to prove Euler's theorem, viz. the fact that every multigraph (i.e. undirected graph without loops but ...
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Proving a graph has an Eulerian path if the components of the graph are both Eulerian.
Suppose that a graph $G$ has a bridge $xy$ such that the components of $G-xy$ are both Eulerian. Prove that G has an Eulerian path. What can you say about the beginning and end of the trail?
I ...
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Eulerian Trail proof in Walk Through Combinatorics
I'm struggling with the proof of Eulerian trail in walk through combinatorics.
As you now, the theorem states that "A connected graph G has a closed Eulerian trail if and only if all vertices of ...
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Is it possible to go over all lines of a grid with a pencil without lifting it or going over a drawn line?
Is it possible to go over all lines of an infinite grid with a pencil without lifting it or going over a drawn line ?
The pencil can cross through a segment already drawn but cannot go over an already ...
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How many ways of traversing every arc of a complete digraph exactly once from a given starting vertex are there?
Given a set of $n$ states $V = \{ s_1, s_2, \ldots, s_n \}$, and a complete digraph $G = (V, A)$ where $A = \{ (a,b) \mid (a,b) \in V^2\; \text{and}\; a \neq b \}$, I'm interested in finding cyclic ...
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"If a vertex appears $k$ times in an eulerian circuit, then it must have degree $2k$" - Why?
I need help understanding this part of this proof from Graphs and Digraphs (7th ed):
Theorem 3.1. A connected multigraph is eulerian if and only if every vertex has
even degree.
The authors of this ...
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Graph theory : Travel to each Edge at least one and returning to the starting Node with the Shortest path in a Weighted Graph?
I would like to travel to each edge of the weighted graph at least once choosing the shortest path, i know this problem is similar to the Chinese Postman Problem CPP, but the difference here is that ...
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Which is correct Euler path or Euler trail?
Since path cannot have repeated vertices, the definition for
A graph which exactly two vertices have odd degree, and all of its vertices with nonzero degree belong to a single connected component
...
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Eulerian graph with $46$ vertices and $45$ edges [closed]
Is $G$ a graph which has an isolated vertex and its other connected component is $C_{45}$ an Eulerian graph? To be an Eulerian graph, could it happen that our graph is not connected?
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How to find a vertex-induced subgraph with Eulerian cycle [closed]
How to find a vertex-induced subgraph with an Eulerian cycle?
The graph is connected and undirected.
Is the problem NP-Hard?
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Networks: Covering each edge and starting and ending at the same vertice when repeating edges must be involved.
I'm in year 11 and my question is regarding a certain topic that I've come across in my curriculum. The problem surrounding this question is about creating a path of minimum length that covers each ...
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Prove that if a graph has an Eulerian path, then the number of odd degree vertices is either 0 or 2
I'm trying to prove that if a graph has an Eulerian path, then the number of odd degree vertices is either 0 or 2.
My attempt.
We know that the sum of the degrees of all vertices is twice the number ...
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When is the partition refinement graph Eulerian?
Let $n$ be a positive integer, and let $p(n)$ be the number of partitions of $n$. For two partitions $p_1, p_2$ of the same integer $n$, we say that $p_2$ is a refinement of $p_1$ if the parts of $p_1$...
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Why is a bipartite graph in which every vertex has degree exactly $2$ is simply a cycle?
I am trying to intuitively understand why a bipartite graph in which every vertex has degree exactly $2$ is simply a cycle. So far, I have tried to intuitively justify this by saying that an Eulerian ...
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Graph Theory - The Mouse problem [closed]
Edward the mouse has just finished his brand new house. The floor plan is shown below:
Edward wants to give a tour of his new pad to a lady-mouse-friend. Is it possible for them to walk through ...
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Confused about how to find de Bruijn sequence from Eulerian tour
I am not following how wikipedia constructs a de Bruijn sequence from an Eulerian tour here. When our Eulerian tour visits vertices $000,000, 001, 011, 111, 111, 110, 101, 011,
110, 100, 001, 010, 101,...
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Are there at least $|V|$ Eulerian tours in $G$ if $G$ is even degree and connected? (My proof)
I believe there to be at least $|V|$ Eulerian tours in $G$ if $G$ is even degree and connected, and want to confirm that my reasoning of this is sound. (I am using the definition of Eulerian tour to ...
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Prove by induction on the length of the walk that whenever it visits a vertex, it has traversed an odd number of edges incident to it
Say we walk on a finite, connected, even-degree graph with no self loops in the following way: we start from an arbitrary vertex $s \in V$, at each step choosing an untraversed edge incident to the ...
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Eulerian Graphs and Cycle Decompositions
I have been trying to find the following references, it would be helpful if I am linked to either of the two, both of them would be ideal.
[1] H. Fleischner, Cycle decompositions, 2-coverings, ...
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Is the following a Path?
While watching a lecture on Eulerian path, I found the following slide
Now my question is how is the above walk a path? The vertex w is getting repeated twice and then $ue_1ve_2we_3xe_4ye_5w$ should ...
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Combinatorics graphs for $2k+1$ representatives from $k $ different countries.
I'm having trouble with the following question :
Representatives from $1+2k$ countries come to an international conference, $k$ representatives from each country.
Is it possible to seat the $k(2k+1)$ ...
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e <= 3v - 6 for planar graph question: why does every face (of a planar graph) have to have at least three sides?
Can't we make a face with just two edges(sides) and two vertices? We just connect those two vertices twice each with different edges and we can make a face between the two edges with only two vertices ...
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Enumerating “Cyclic Double Permutations”
This is a generalization of a question first asked by loopy walt on Puzzling Stack Exchange: https://puzzling.stackexchange.com/q/120243. I asked the following version of the question in the comments, ...
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Understanding which conditions a graph has a Eulerian Path
The graph $Q_n$ is a graph with 2n vertices, where each vertex is associated with a string of 1's
and 0's of length n, and where two vertices are adjacent if and only if their associated strings
...