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Questions tagged [hamiltonian-path]

A path in a graph that visits each vertex exactly once.

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Are Euler trails and tours of a graph the same as Hamilton paths and cycles of the corresponding “edge graph”?

Is knowing information about Euler paths and Euler tours about a graph $G$ the same as knowing information about Hamilton paths and cycles of the graph $H$ obtained from $G$ such that vertices of $H$ ...
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38 views
+50

How to calculate quantity of Hamilton cycles

If I have $n^2$ vertices and each vertex is adjacent to $2n-2$ vertices, how may I calculate the quantity of possible Hamilton cycles? Would I need to modify the computation if I stipulated that I'm ...
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Graph which is Bipartite, has an Euler circuit, but not a Hamiltonian circuit

Is there a graph which is bipartite, has an Euler circuit, but not a Hamiltonian circuit? I know the answer is yes, but if you consider something like this: I don't think this would be bipartite, ...
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31 views

Prove that the connected circulant regular graphs of degree at least three contain all even cycles.

This is the question I am trying to solve, but while researching about circulant graph I came across Paley's graph of order 13. Now clearly when looking at this graph which is an example of circulant ...
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Would removing the max weighted edge from a Hamiltonian circuit result in a Hamiltonian path?

If I have a Hamiltonian circuit, assuming that cutting the max weighted edge resulted in a Hamiltonian path. Is it guaranteed that the path would be minimal too?
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183 views

Show that the line graph of any quasi-cyclic graph contains a Hamiltonian cycle.

The definition I have been given for a quasi-cyclic graph is as follows: a graph $G=(V, E)$ is quasi-cyclic if $1)$ it contains a unique cycle $C=(V(C), E(C))$ and $2)$ for each edge $xy$ in $E$ at ...
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30 views

Finding number of two cycle multi-graphs [closed]

Consider directed graphs on n labelled vertices {1,2,...n}, where each vertex has exactly one edge coming in and exactly one edge going out. We allow self-loops. How many such graphs have exactly two ...
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Number of Hamiltonian cycles in a random graph

I want to show that $\mu_{n}(p)$, the expected number of Hamiltonian cycles in the random graph $G(n,p)$, is given by $$\mu_{n}(p)=\frac{1}{2}(n-1)!p^{n}$$ We can easily show that the number of the ...
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35 views

Limiting behavior of the expected number of Hamiltonian cycles in the random graph $G(n, p)$.

So we have that the expected number of Hamiltonian cycles in the random graph $G(n,p)$ is given by: $\mu_{n}(p)=\frac{1}{2}(n-1)!p^{n}$ for $n \geq 3$. We now want to find lim$_{n\to \infty}\mu_{n}(...
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Combinatorial proof of Hamiltonian paths on the rook graph

We can be sure that number of Hamiltonian paths on the rook graph for any single cell on $n\times2$ chessboard equals $$ H(n+1) = \sum_{k=0}^{n} \binom{n}{k} \binom{k}{\lfloor{\...
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1answer
23 views

What is a vertex-transitive graph? (Question about Lovász Conjecture)

I was reading about Lovász Conjecture and came across the following definition on Wikipedia of a vertex-transitive graph (see below). $\bullet$ It states that a graph is vertex-transitive if for any ...
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Hamiltonian circuit

Prove that a graph that posses a Hamiltonian circuit must have no pendant vertices. To prove this, each vertex in a graph, that also has a hamiltonian circuit, much acquire at least two edges in ...
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Hamiltonian paths and cycles of rook graph on $n\times2$ chessboard

According to OEIS, there are closed form for directed Hamiltonian paths (A096121) and Hamiltonian cycles (A276356) of rook graph on $n\times2$ chessboard. Are there papers which include proof of those ...
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1answer
32 views

Number of different graphs with this degree sequence

The set of degree sequences in question is: $$ D_1=\{4^4,6^4,4^4\} $$ $$ D_2=\{4^4,6^4,6^4,4^4\} $$ $$ D_3=\{4^4,6^4,6^4,6^4,4^4\} $$ $$ D_4=\{4^4,6^4,6^4,6^4,6^4,4^4\} $$ $$ ... $$ $$ D_N=\{4^4,...
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Path / Graph problem with X nodes looking for Y paths with the most similar length.

I have the following graph / path problem: There is exactly 1 start node and 1 end node. There are also X (in this case 7) nodes, each connected to all other nodes and the start and end node with ...
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1answer
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Graph Problem: find X path lengths which are closest to each other (similar to shortest path)

I have the following problem: 1 Start point 1 End point 7 Knots / waypoints which you have to visit in any order All 9 knots are connected with each other. All paths have to start at the start point ...
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is it possible to move a knight on a chessboard such that it completes every permissible move exactly once?

a move between two squares is counted as one regardless of the direction. basically, we want to prove that a knight started from any position in a 8*8 chessboard can go to all the possible places in ...
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1answer
36 views

What is the term for a graph in which each edge belongs to a Hamiltonian cycle?

Furthermore, are they any known results about these graphs, such as necessary or sufficient conditions for a graph to have this property?
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About the proof of a graph is not Hamiltonian.

Given the following graph: https://i.stack.imgur.com/fg2Q9.png Is this graph Hamiltonian or not? The answer is no. What I tried to prove is by using the fact that: "if a vertex in the graph has ...
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1answer
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Show that if $|G| \geq 5$ and for each pair of vertices $u,v$ there is an $u−v$ Hamilton path then $\kappa(G) \geq 3$

Show that if $G$ is a graph with $|G| \geq 5$ such that for each pair of vertices $u,v$ there is an $u−v$ Hamilton path in $G$, then $\kappa(G) \geq 3$ I do not really have any idea how to start this ...
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Prove that if a graph $G$ has a Hamilton path, then for every $S\subseteq V(G)$, the number of components of $G-S$ is at most $|S|+1$.

Prove that if a graph $G$ has a Hamilton path, then for every $S\subseteq V(G)$, the number of components of $G-S$ is at most $|S|+1$. My solution (rough and incorrect): Consider a Hamilton path $P$ ...
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0answers
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Number of open and closed rook's tours

Knight's tour is very well known problem, but what about rook's tour? On $n\times1$ chessboard there are obviously $n!$ open and $(n-1)!$ closed tours. Is there a way to easily compute number of open ...
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Hamiltonian paths in graph

I have a theorem about Hamiltonian paths in graph, but I doubt it's possible. Theorem If vertex $v$ of graph $G$ is not isolated, and degree of every other vertex is $\geq k$ for $k \geq 2$, and $|V| ...
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Every self-complementary graph contains a Hamiltonian path.

How to show that every self-complementary graph is traceable (contains a Hamiltonian path)? Definitions: Self-complementary graph Hamiltonian-Path Traceable Graph
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Diameter of a graph such that given independent vertices u, v, d(u) + d(v) ≥ n.

given my continuous struggle with proofs on graph theory, I come with another problem I do not know how to approach. Given a graph G = (V, E) such that for any two non-neighboring vertices u, v ∈ V ,...
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Is this graph Hamiltonian and how to prove it is not? [closed]

How to prove this graph is not Hamiltonian (does not contain a Hamiltonian cycle)? I have already tried removing some vertices from the graph, but I cannot get more than 7 connected components after ...
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17 views

Hamiltonian cycles and paths in a graph

I want to present you a lemma, that I've almost proved, but i'm stuck at the very end of it. Lemma If vertex $v$ of a graph $G$ is not isolated and degree of every vertex except $v$ is $\geq k$ (for $...
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1answer
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All 4-connected planar graphs are Hamiltonian-connected

I started reading Thomassen's paper A Theorem on Paths in Planar Graphs, where he proves one of Plummer's conjectures: Every $4$-connected planar graph is Hamiltonian-connected. Context. Recall that ...
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37 views

Hamilton Paths in Complete graph $K_n$

In complete graph $K_n$, is it true that we can have at least $2*n$ Hamilton paths?
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Hamilton path and Euler circuits in Cycle graph

Can $C_n$ has $2*n$ Euler circuits and $3*n$ Hamilton paths in the Cyclic graphs?
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A Hamilton graph having a Hamilton cycle that traverse an edge more than once.

I was asked to draw a Hamilton graph having a Hamilton cycle that traverse an edge more than once. My first impression of this question was: what? I mean if we are not allowed to visit a vertex more ...
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1answer
57 views

Is there a non-planar, non-hamiltonian and eulerian graph?

I'm trying to find a graph that is non-planar, non-hamiltonian and eulerian but I can't find anyone. Is this possible? Thanks
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62 views

How to show that complement a of regular graph is a Hamiltonian graph? [closed]

I have a regular graph G of degree k ≥ 1 (ie its every vertex is of degree k) with at least 2k+2 vertices. How do I show that complement of G is a Hamiltonian graph?
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How many Hamiltonian cycles in a complete graph cover edges that don't share vertices?

Consider a complete graph, K, that has n vertices. There is a set of edges within K that have a common property, which is that they do not share vertices anywhere on the graph. Let's call these set ...
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How many Hamiltonian cycles are there in a complete graph that must contain certain edges?

Consider a complete graph $G$ that has $n \geq 4$ vertices. Each vertex in this graph is indexed $[n]=\{1,2,3, \dots n\}$ In this context, a Hamiltonian cycle is defined solely by the collection of ...
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How many Hamiltonian cycles are there in a complete graph if we discount the cycle's orientation or starting point?

Consider a complete graph G with n vertices. Each vertex is indexed by [n] = {1,2,3...n} where n >= 4. In this case, a Hamiltonian cycle is determined only by the collection of edges it contains, ...
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1answer
40 views

Hamiltonian paths in a simple graph

If a simple graph $G$ with $n$ vertices has a Hamiltonian cycle, what can we say about the number of Hamiltonian paths that $G$ has? Since Hamiltonian cycle goes through each vertex only once the ...
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1answer
127 views

There exists a graph on $n$ vertices such that every vertex has degree at least $\frac{1}{2}n -1$

Show that for every $n \geq 1$ there exists a graph on $n$ vertices such that every vertex has degree at least $\frac{1}{2}n -1$ and G is not Hamiltonian. I know that Dirac's theorem implies every ...
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1answer
111 views

Hamiltonian Knight's (closed) walk for odd $\times$ odd chess board

I am taking a course on graph theory right now and we were posed the following question: Show that if $n$ is odd, a knight on an $n \times n$ chessboard can not make a closed tour of the chessboard ...
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255 views

Number of Hamiltonian cycles in complete graph Kn with constraints

I am currently working on a exercice which aims to count the number of hamiltonian cycles in a complete graph. Since it is a completely new topic to me, I struggle to think about how to solve the ...
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31 views

Difference between hamiltonian and pre-hamiltonian path?

What is the difference between a hamiltonian path and a pre-hamiltonian path? Or it is the same? How do I show that a digraph G contains a pre-hamiltonian path?
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1answer
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Need prove a Graphs with n >= 5 with a Euler path and without Hamilton Path and Hamilton Cycle exist.

I already proved a Euler Tour can't exist because all degrees would have to equal a number divisible by 2 but a Euler Path requires two odd degrees. I still have to prove a graph exists that contains ...
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Pseudo hamiltonian connected property of a graph

Is there a connection between pseudo hamiltonian connectedness and hamiltonicity of graphs?
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Prove a lower limit of $|E(G)|$ where for any $u,v\in G$, there exists a Hamilton path

Define a "Hamilton-connected graph $G$" as For any vertices $u, v \in G$, a Hamilton path exists, where the two ends of the path are vertices $u$ and $v$. Try to prove that if $G$ is a Hamilton-...
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54 views

$G$ has a Hamiltonian path iff $G+v$ has a Hamiltonian cycle

If $G = (V, E)$ is a simple graph with at least one vertex and $G'$ is the graph formed by adding a new vertex $v$ and making it adjacent to every vertex in $V$. How do you show that $G$ has a ...
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2answers
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Where is the proof of Tutte's graph having no Hamiltonian cycles?

Tutte's graph was/is a famous counterexample to Tait's conjecture that every cubic, polyhedral graph has a Hamiltonian cycle. However, I cannot get access to Tutte's original paper (it's stuck behind ...
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The number of Hamiltonian paths in a tournament is always odd?

A tournament is defined as an orientation of a complete graph. Prove or disprove the following statement: In a tournament, there are exactly an odd number of Hamiltonian paths. In all cases I’ve ...
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70 views

if G does not have vertices of odd degree, then there are disjoints cycles by edges

Show that if G does not have vertices of odd degree, then there are disjoints cycles by edges $C_{1}, C_{2}, C_{3},...C_{m}$ such that $E(G)=E(C_{1}) \cup E(C_{2})\cup ...\cup...\cup E(C_{M})$ I ...
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Is there a Hamiltonian path? (i.e. can the general associative law be solved with Graph Theory?) [duplicate]

Consider the different bracketings of the summation $$1+2+3+4+5\,.$$ I've listed them all below: $$ 1+(2+(3+(4+5)))\,,\quad (1+((2+3)+4))+5\\ 1+(2+((3+4)+5))\,,\quad (1+(2+(3+4)))+5\\ 1+((2+(3+4))+...
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Doubt on the definition of closure of a graph.

The closure of a graph $G$, denoted $cl(G)$ is defined to be the supergraph of $G$ obtained from $G$ by recursively joining pairs of nonadjecent vertices whose degree sum is atleast $n$ untill no ...