Questions tagged [hamiltonian-path]

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

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Graph theory: decompositions and Hamiltonian graph

13 people who are not superstitious wish to have dinner together at a round table for a few nights so that each person has different neighbours every night. For how many nights can they do this? ...
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Show that undirected connected 3-regular graph with 8 vertices has Hamiltonian path

Let be $G=\langle V,E\rangle$ undirected and connected 3-regular graph with 8 vertices. Prove that $G$ has Hamiltonian path. I am trying to prove the above claim, however I don`t know how to develop ...
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Hamiltonian path in $S_n$?

Say $S_n$ is the symmetric group. Define a graph $G$ by $G=(S_n,E)$, where there is an edge from $\sigma_1$ to $\sigma_2$ if and only if $\sigma_2=t\sigma_1$ for some transposition $t$. Is there a ...
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42 views

Show that $G$ is not Hamiltonian.

Suppose that $G$ is a graph with $pq$ vertices where $p$ and $q$ are primes and $2<p<q$. If $(p-1)(q-1)$ vertices of $G$ have degree $p+q-1$ and all other vertices have degree $pq-1$, show ...
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Algorithm to construct $k$ edge-disjoint Hamiltonian cycles

Consider an undirected graph $G=(V,E)$, where $V=\{1,2,\ldots, n\}$, and initially $E=\varnothing$. Now take the following steps: In the $1$st round, add undirected edges $(1,2)$, $(2,3)$, $\ldots$, $...
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Hamilton cycle problem on group's Cayley graph. Determining a function $G^n \to \Bbb{Z}[G]$.

Let $G$ be a finite group and $G^n$ be the direct product of $n$ copies of $G$. Let $f_g: G^n \to \Bbb{Z}, \forall \ g\in G$ be such that there exists at least one $x \in G^n$ such that $f_g(x) = 1, \...
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A Hamiltonian cycle generated by lame rooks moves

I have got this problem at high-school math-contest seminar on Graph Theory Let us have a chessboard, where one black and one white lame rooks stand. Lame rook can move to edge-adjacent field only. ...
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Hamilton Cycle Proof Verification

Explain why there is no Hamilton cycle in the graph attached below. Then, show that if any pair of non-adjacent vertices in the same graph are joined by an edge, then the resulting graph has a ...
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$n^{th}$ Dimensional Prime Nexus Conjecture (Hamilton's Path & Cycle)

The system and proposition of Prime Nexus - We have to arrange $n>1$ Distinct natural numbers in a sequence such that the adjacent elements sums upto any Prime number. Convenience is when we set ...
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Travelling salesman problem visiting different nodes different times [closed]

Hi I am trying to solve a more complicated travelling salesman problem (shortest path visiting all nodes in a directed graph), where (1) I need to revisit different nodes for different times, (2) I ...
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How to find the number of cycles and set of nodes in each cycle in an undirected, connected and loopless graph?

This question could be repetetive. I tried to look up for some posts, most of them are to check whether a graph contains a cycle or not. Assume there is no multiple edge. My problem is I need to find ...
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Induction Proof On “Tree with Nodes as Cycles” Graph

So I have one question about defining the type of graph I was working on: Define A Graph - Tree Graph With "Cycles" as Nodes A short summary for the graph I would like to define as G=(V,E): ...
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Define A Graph - Tree Graph With “Cycles” as Nodes

I am working on my thesis and I would like to have a proper definition for this type of graph: "Tree Graph With Nodes As Cycles" I would like to define a graph similar to a "simple tree", but some of ...
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Existence of Hamiltonian Path using minimum and maximum degrees

In a simple graph $G$ with $n$ nodes, let $c<\deg(V)<C \space \forall$ nodes $V$, for given constants $c<C<{n}$. Let the total number of edges in $G$ be $E$. What is the maximum value of $...
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Efficient hamiltonian path through neighboring squares

My knowledge of Number/Graph theory is very limited. I'm sorry if someone can find this answer posted elsewhere quickly, I spent some time searching but don't know enough to know what to search. The ...
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Symmetric difference of Hamilton circuits in planar cubic graphs

The answer to math.stackexchange.com/questions/3235317/every-cubic-3-connected-hamiltonain-graph-has-three-hamiltonian-cycles-with-spec?rq=1 points out that the cube graph contains no three Hamilton ...
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double tours embedding of nonhamiltonian bicubic graphs

Can Georges Graph (or any other nonhamiltonian bicubic graph ) be embedded on an oriented surface of genus -2, i.e. a double torus? If it helps, it would have $F=E+\chi-V=75-2-50=23$ faces...
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Are Hamilton Cycles Petrie Polygons w.r.t. the Local Orientation?

Consider planar cubic bipartite graphs. The graph has a 3-edge coloring due to the 4-coloring theorem. By that and its planarity the vertices have an induced orientation. Now traverse the graph's (...
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Hamilton path reduction to Hamilton cycle

HAMILTON PATH: given a directed graph $G$ and $2$ nodes start and end does there exist a hamilton path from start to end? HAMILTON CYCLE: given a directed graph $G$ and $1$ node start, does there ...
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Does knowing a graph has a Hamiltonian Cycle make it easier to find the cycle?

Given a simple and connected graph $G=(V, E)$. I know it's NP-Complete to determine if $G$ has a Hamiltonian Cycle (HC). But if we know $G$ indeed contains an HC, can we find the cycle in poly-time?
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Why is hamiltonian path reduction to cycle wrong

Wikipedia link states that you are able to reduce HP to HC by adding a single vertex that is connected with all the edges. I understand the reason why the reduction from cycle to path works by adding ...
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Hamiltonian paths in cubic graphs with cyclically edge connectivity at least 5

Definitions A cubic graph (simple) $G$ is a 3-regular graph. An edge cut $K$ is cycle separating if $G-K$ is disconnected and at least two components of $G-K$ have circuits. A graph is cyclically ...
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NP-Complete proof of deciding if a graph has another Hamiltonian Circuit

I need to prove as an exercise that the following problem is NP-Complete: Given a graph and an already existing Hamiltonian Circuit in that graph, decide if the graph has another Hamiltonian Circuit ...
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$G$ contains Hamiltonian circuit $\Leftrightarrow G + uv$ contains Hamiltonian circuit

We have graph $G$ and two not connected vertexes $u,v$ where $$ \deg(u)+\deg(v) \ge n $$ Prove that $G$ contains Hamiltonian circuit $\Leftrightarrow G + uv$ contains Hamiltonian circuit My ...
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Finding a Hamiltonian Cycle from a perfect matching on a the bipartite graph

A disjoint vertex cycle cover can be found by a perfect matching on the bipartite graph constructed from the original graph (L) and its copy (R) and with L original graph edges replaced by ...
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How do we know that the prime sum graph defined in this paper has a perfect matching?

https://arxiv.org/pdf/1804.07104.pdf I am reading this paper and trying to understand why Theorem 1.1 guarantees a perfect matching for the prime sum graph as stated at the bottom of page 2. Can ...
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Determine the minimum number of robot teams

I have completed parts $a$ and $b$, but I am stuck on how to show my work for part $c$. I believe the minimum number of robot teams is $3$, but I'm not sure how I got that answer. How do I start this ...
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Traveling sales man with alternating groups of destinations.

I am looking for an algorithm to determine the best route where the groups of destinations alternate. For example: I have three stores and three warehouses. So I would like to visit a warehouse first,...
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Playing Doublets with the Primes

Lewis Carroll's famous game of Doublets is well known. In it you are asked to transform a given word into another by changing only one letter at a time, forming a genuine new word (not a proper name) ...
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Determine if a Hamiltonian Cycle exists?

Suppose I have a graph $G=(V, E)$. Removing a subset $R$ of edges from $G$ results in a new graph $G^\prime=(V, E\setminus R)$. The maximum number of edges in $R$ is $|E|$. Suppose I have the graph $G^...
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How to show that there exists a tournament of size n with a certain number of Hamiltonian paths? [duplicate]

Assuming a tournament of size $n$ is a choice of orientation for each edge of $K_n$, show that there exists a tournament of size $n$ with at least $\dfrac{n!}{2^{n-1}}$ Hamiltonian paths. This ...
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Hamiltonian Cycles of a graph

Prove if this statement is true: every graph consisting of two edge-disjoint Hamiltonian paths contains a Hamiltonian cycle. Two edge-disjoint Hamiltonian paths means all vertices can be connected ...
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Proof that if graph has $\frac{(n-1)(n-2)}{2} + 2$ edged then contains hamiltonian cycle

Proof that if graph has $\frac{(n-1)(n-2)}{2} + 2$ edged then contains hamiltonian cycle I think that it is good to use there induction: Let check base of induction. For $ n= 3 $ I have $$ |E| = \...
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Chance of not finding an alternating Hamiltonian path in a colored graph in $n$ random walks

Introduction Given a random graph which can be constructed as follows: generate $s$ pairs of unconnected black edges and $2 \cdot s$ vertices connect these black edges using red edges, where the ...
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Short explanation for hamiltonian cycles in $Q_3$

The task is to find the count of hamiltonian cycle in $Q_3$. So I know the answer is $6$, but i don't know why or how to get it. My first attempt was simple: I start from edge $6$, and I have $3$ $(...
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Finite Graph With Degree $\geq 2$

Show that a finite graph with all vertices with degree $\geq 2$ has a cycle that contains a vertex which is non-adjacent to any other vertices not contained in the cycle. I tried to start at an ...
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Hamiltonian Cycle with n vertex graph

Let a n-vertex graph such that every pair of not adjacent vertices a & b has degree(x) + degree(y) $\geq$ n. Show the graph contains a Hamiltonian cycle. By dirac's thm, a simple graph with n ...
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Reduce Hamiltonian Path Decision Problem To Hamiltonian Cycle Decision Problem

Person A requires that he determine whether or not a particular graph G = (V,E) has a Hamiltonian path from vertex a to vertex b. His colleague Person B has implemented a function that takes an ...
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Proof of equivalence of S-TSP solution with TSP solution (metric instances)

I am wondering where could I find proof for following S-TSP to TSP transformation. S-TSP (Steiner Travelling Salesman Problem) def: Let $G=(V, E)$ be a non-directed weighted graph. Let $V' \subset V$ ...
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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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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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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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How to convert a maximal planar graph to a regular planar multigraph?

Given a maximal planar graph (coming from the convex hull of a set of points on a sphere), I want to add edges until it is regular (all vertices touch the same number of edges), while keeping it ...
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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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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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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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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{\...