Prove that every $8$-regular graph has $4$- and $2$-regular spanning subgraphs. Note: A graph is spanning subgraph, if it contains every vertex of the original graph. Furthermore this example’s from Hamiltonian cycle/path topic.

I am trying to figure out if a graph can be assumed Hamiltonian or not, or if it’s indeterminable with minimal information: A graph has 17 vertices and 129 edges. Hamiltonian graphs are proved by, as long as the vertices are greater than or equal to 3 (which this is), then if the sum of […]

I need to show that the maximum number of edges of non-Hamiltonian, simple graph, on $n$ vertices noted by $t(n,H_n)$ is $\binom{n-1}{2} + 1$. It’s essential to show the upper and lower bounds for this question. Now, I can see why I need to show the upper bound – for $t(n,H_n) > \binom{n-1}{2} + 1$ […]

How to prove “Prove $P_m X P_n$ (graph cartesian product) is Hamiltonian if and only if at least one of $m,n$ is even” Graph cartesian product is Grid graph, if I figure out $P_2 X P_2$ it will be $C_4$ (it is Hamiltonian). How to state this prove ?

I’m trying to find out if it is possible to construct a connected Hamiltonian and a connected non-Hamiltonian graph using the same degree sequence. For disconnected graphs it would be easier, I could choose a degree sequence of $(2,2,2,2,2,2)$ and have $C_6$, which has a Hamiltonian path, and two disjoint 3-cycles, which are non-Hamiltonian. How […]

It is generally difficult to determine whether a (large) graph have no Hamilton cycle (As opposed to determining whether it has any Euler circuit). This example illustrates a method (which sometimes work) to indicate that a graph has no Hamilton cycle. a. Show that if m and n are odd integers (not both = 1), […]

For any graph with order $n \geq 3$, given that its size is $$m \geq \frac{\left(n-1\right)(n-2)}{2} + 2,$$ show that the graph is Hamiltonian. I know that if I can show that the degree sum of any two non-adjacent vertices is $\geq n$, then I’d be done. Likewise, if I could show that the above […]

In a complete graph with $n$ vertices there are $\frac{n−1}{2}$ edge-disjoint Hamiltonian cycles if $n$ is an odd number and $n\ge 3$. What if $n$ is an even number?

The knight’s tour is a sequence of 64 squares on a chess board, where each square is visted once, and each subsequent square can be reached from the previous by a knight’s move. Tours can be cyclic, if the last square is a knight’s move away from the first, and acyclic otherwise. There are several […]

Intereting Posts

Subspaces of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?
How to prove that a polynomial of degree $n$ has at most $n$ roots?
Is $\mathbb{Z}\over \langle x+3\rangle$ field?
What is the $\lim_{n\to \infty} {\sqrt{(1+1/n)(1+2/n)…(1+n/n)}}$
Quantifiers, predicates, logical equivalence
Do these matrix rings have non-zero elements that are neither units nor zero divisors?
Generalizing values which Euler's-totient function does not take
How to prove that $\frac{x^2}{yz+2}+\frac{y^2}{zx+2}+\frac{z^2}{xy+2}\geq \frac{x+y+z}{3}$ holds for any $(x,y,z)\in^3$
Inequality Of Four Variables
Second order ODE – why the extra X for the solution?
Definition of local maxima, local minima
Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$.
Prove $6 \nmid {28} – 3 \right)^{-n}]$
How to evalutate this exponential integral
Finding a unique subfield of $\mathbb{Q}(\zeta)$ of degree $2$?