Intereting Posts

Matrix Equation $A^3-3A=\begin{pmatrix}-7 & -9\\ 3 & 2\end{pmatrix}$
Why is a random variable called so despite being a function?
Real integral by keyhole contour
little-o and its properties
Generating Coprime Integers
the purpose of induction
Diophantine equation $a^m + b^m = c^n$ ($m, n$ coprime)
polynomial approximation in Hardy space $H^\infty$
do all uncountable sets have same cardinality as real numbers?
Questions on atoms of a measure
“Random” generation of rotation matrices
Improper integrals with u-substitution
Quantification over the set(?) of predicates
If $p\geq 5$ is a prime number, show that $p^2+2$ is composite.
The categories Set and Ens

Given a cubic planar hamiltonian graph with $F$ faces. Let $a_k$ be the number of face of degree $k$ inside and $b_k$ outside the Hamilton cycle. We have the following:

- $\sum \limits_k \left(a_k+b_k\right)k = 6F-12$ (due to Euler; my attempt)
- $\sum \limits_k \left(a_k-b_k\right)(k-2)=0$ (Grinberg’s Theorem)

Are these two all of this type or are there more?

- Can any finite group be realized as the automorphism group of a directed acyclic graph?
- On the eigenvalues of “almost” complete graph ?!
- First book on algebraic graph theory?
- associativity in graph theory
- Complexity of counting the number of triangles of a graph
- suppose $n$ people are in a party and every two of them have exactly one common friends,prove that there is one who is friend to all.

- Constructing self-complementary graphs
- to clear doubt about basic definition in graph theory
- A less challenging trivia problem
- isomorphic planar graph with its complement
- Name this object derived from the Rhombicuboctahedron…
- 3-regular graphs with no bridges
- If $G$ is an acyclic graph, How can we prove that $G$ is connected?
- How to count the no. of distinct ways in which $1,2,…,6$ can be assigned to $6$ faces of a cube?
- (Olympiad) Minimum number of pairs of friends.
- Number of vertices of a complete graph with $n$ edges

Let’s restrict it further to bipartite, cubic, hamiltonian Graphs $G$ only made up of $4$- or $6$-faces. These graphs can build out of $6$ faces of degree $4$ and the rest of degree $6$. Let $F$ be the number of faces, $V$ the number of Vertices, $E$ the number of edges and $a_k$ be the number of face of degree $k$ inside and $b_k$ outside the Hamilton cycle. Starting from Euler, we get:

$$

F+V=E+2\\

F=\sum_{k\in \{4,6\}} a_k+b_k = E-V+2

$$

For $3$-regular graphs, we have $2E=3V$ and $a_4+b_4=6$ so

$$

6+a_6+b_6= V\left(\frac{3}2-1\right)+2 \\

2(a_6+b_6)= V-8\tag{1}

$$

By combining $(1)$ and

$$a_4+b_4=6\tag{2}$$ with Grinberg’s formula (divided by $2$) in the given case

$$

(a_4-b_4)+2(a_6-b_6)=0\tag{3}

$$

we arrive at

- $$

a_4+2a_6=\frac12V-1 \tag{$\frac12[(1)+(2)+(3)]$}

$$

I checked some examples and it seems to work. Tell me if there is anything wrong or unclear.

Depending on $\frac V2$ being even or odd there are $1,3,5$ or $(0,)2,4,6$ possible $4$-faces inside the HC, where I’m not sure if $0$ is possible…

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