which complete graphs are hamiltonian ?

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which complete graphs are hamiltonian ?

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https://en.wikipedia.org/wiki/Hamiltonian_path

https://en.wikipedia.org/wiki/Hamiltonian_path
Hamiltonian path – Wikipedia
As complete graphs are Hamiltonian, all graphs whose closure is complete are Hamiltonian, which is the content of the following earlier theorems by Dirac and Ore. Dirac’s Theorem (1952) — A simple graph with n vertices ( n ≥ 3 {displaystyle ngeq 3} ) is Hamiltonian if every vertex has degree n 2 {displaystyle {tfrac {n}{2}}} or greater.

https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/12:_Graph_Theory/12.08:_Hamilton_Cycles

https://math.libretexts.org/Bookshelves/Applied_Mathematics/Contemporary_Mathematics_(OpenStax)/12:_Graph_Theory/12.08:_Hamilton_Cycles
12.8: Hamilton Cycles – Mathematics LibreTexts
Counting Hamilton Cycles in Complete Graphs. Now, let’s get back to answering the question of how many Hamilton cycles are in a complete graph. In Table 12.8, we have drawn all the four cycles in a complete graph with four vertices. Remember, cycles can be named starting with any vertex in the cycle, but we will name them starting with vertex …
Hamiltonian Circuits
Recall the way to find out how many Hamilton circuits this complete graph has. The complete graph above has four vertices, so the number of Hamilton circuits is: (N – 1)! = (4 – 1)! = 3! = 3*2*1 = 6 Hamilton circuits. However, three of those Hamilton circuits are the same circuit going the opposite direction (the mirror image).

https://math.libretexts.org / Bookshelves / Applied_Mathematics / Book:_College_Mathematics_for_Everyday_Life_(Inigo_et_al) / 06:_Graph_Theory / 6.04:_Hamiltonian_Circuits

https://math.libretexts.org / Bookshelves / Applied_Mathematics / Book:_College_Mathematics_for_Everyday_Life_(Inigo_et_al) / 06:_Graph_Theory / 6.04:_Hamiltonian_Circuits
6.4: Hamiltonian Circuits – Mathematics LibreTexts
Recall the way to find out how many Hamilton circuits this complete graph has. The complete graph above has four vertices, so the number of Hamilton circuits is: (N – 1)! = (4 – 1)! = 3! = 3*2*1 = 6 Hamilton circuits. However, three of those Hamilton circuits are the same circuit going the opposite direction (the mirror image).

https://mathworld.wolfram.com/HamiltonianGraph.html

https://mathworld.wolfram.com/HamiltonianGraph.html
Hamiltonian Graph — from Wolfram MathWorld
A Hamiltonian graph, also called a Hamilton graph, is a graph possessing a Hamiltonian cycle. A graph that is not Hamiltonian is said to be nonhamiltonian. A Hamiltonian graph on n nodes has graph circumference n. A graph possessing exactly one Hamiltonian cycle is known as a uniquely . While it would be easy to make a general definition of that considers the …

https://math.mit.edu/~fgotti/docs/Courses/Combinatorial Analysis/22. Hamiltonian Graphs/Hamiltonian Graphs.pdf

https://math.mit.edu/~fgotti/docs/Courses/Combinatorial Analysis/22. Hamiltonian Graphs/Hamiltonian Graphs.pdf
PDF Lecture 22: Hamiltonian Cycles and Paths – MIT Mathematics
Then cycles are Hamiltonian graphs. Example 3. The complete graph K n is Hamiltonian if and only if n 3. The following proposition provides a condition under which we can always guarantee that a graph is Hamiltonian. Proposition 4. Fix n 2N with n 3, and let G = (V;E) be a simple graph with jVj n. If degv n=2 for all v 2V, then G is …

https://www2.math.uu.se/~andersj/graphtheory/lec6.pdf

https://www2.math.uu.se/~andersj/graphtheory/lec6.pdf
PDF Lecture 6: Hamiltonian graphs
Independent sets of Hamiltonian graphs Let Gbe a graph with independent set SˆV. An independent set must not take up to many edges for the graph to be Hamiltonian. … A tournament on nvertices is an orientation Tof the complete graph K n. 1.Show that there is a dircteed Hamilton path in T. (Redei’s theorem) Diracs theorem

https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Combinatorics_(Keller_and_Trotter)/05:_Graph_Theory/5.03:_Eulerian_and_Hamiltonian_Graphs

https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Applied_Combinatorics_(Keller_and_Trotter)/05:_Graph_Theory/5.03:_Eulerian_and_Hamiltonian_Graphs
5.3: Eulerian and Hamiltonian Graphs – Mathematics LibreTexts
A graph G = (V, E) G = ( V, E) is said to be hamiltonian if there exists a sequence (x1,x2, …,xn) ( x 1, x 2, …, x n) so that. Such a sequence of vertices is called a hamiltonian cycle. The first graph shown in Figure 5.16 both eulerian and hamiltonian. The second is hamiltonian but not eulerian. Figure 5.16.

https://www2.math.uu.se/~andersj/graphtheory/lec7.pdf

https://www2.math.uu.se/~andersj/graphtheory/lec7.pdf
PDF Lecture 7: Hamiltonian graphs and colourings – Uppsala University
Independent sets of Hamiltonian graphs Let Gbe a graph with indepen-dent set SˆV. An independent set must not take up to many edges for the graph to be Hamiltonian. … A tournament on nvertices is an orientation Tof the complete graph K n. 1. 1.Show that there is a dircteed Hamilton path in T. (Redei’s theorem) Diracs theorem Theorem 1. Given …

https://mathworld.wolfram.com/HamiltonianCycle.html

https://mathworld.wolfram.com/HamiltonianCycle.html
Hamiltonian Cycle — from Wolfram MathWorld
Il y a 5 joursA Hamiltonian cycle, also called a Hamiltonian circuit, Hamilton cycle, or Hamilton circuit, is a graph cycle (i.e., closed loop) through a graph that visits each node exactly once (Skiena 1990, p. 196). A graph possessing a Hamiltonian cycle is said to be a Hamiltonian graph. By convention, the singleton K_1 is considered to be even though it does not posses a …

https://mathspace.co/textbooks/syllabuses/Syllabus-1030/topics/Topic-20297/subtopics/Subtopic-266708

https://mathspace.co/textbooks/syllabuses/Syllabus-1030/topics/Topic-20297/subtopics/Subtopic-266708
4.05 Eulerian and Hamiltonian graphs | Year 12 Maths – Mathspace
Hamiltonian paths in complete graphs. The following graph is a complete graph. Recall that means each vertex is connected to every other vertex. It has many semi-Hamiltonian paths and Hamiltonian cycles. How many can you find? Remember, a semi-Hamiltonian path is a path where we visit each of the vertices only once.
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