Intereting Posts

How to prove that $\sum _{k=0}^{2n-1} \frac{(-2n)^k}{k!}<0 $
Suppose an entire function $f$ is real if and only if $z$ is real. Prove that $f$ has at most $1$ zero.
$u$ is a $C^2$ solution of $u_t – \Delta u = f(u)$ and $u=0$ on $\partial\Omega \times (0,\infty)$. Show if $u(x,0) \geq 0$, then $u(x,t) \geq 0$
Modular congruence, splitting a modulo
Sum : $\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)^3}$
How to properly apply the Lie Series
Show that the parameterized curve is a periodic solution to the system of nonlinear equations
Newton's method for square roots 'jumps' through the continued fraction convergents
Why is minimizing least squares equivalent to finding the projection matrix $\hat{x}=A^Tb(A^TA)^{-1}$?
Limit of ${x^{x^x}}$ as $x\to 0^+$
Are functions of independent variables also independent?
Review of my T-shirt design
Period of 3 implies chaos
Prove Laurent Series Expansion is Unique
What should be the characteristic polynomial for $A^{-1}$ and adj$A$ if the characteristic polynomial of $A$ be given?

Hamiltonicity remains NP-complete for 2-vertex-connected cubic planar bipartite graphs.

What is the smallest 2- or 3-vertex-connected cubic planar bipartite graph with only one (forward and backward counted as one) Hamilton cycle (HC)?

I thought asymmetry might help, but even Frucht’s graph, a non-bipartite example, seems to have more than one HC…

$\phantom{}$

- Rank of an interesting matrix
- Is there a reason why the number of non-isomorphic graphs with $v=4$ is odd?
- For a Planar Graph, Find the Algorithm that Constructs A Cycle Basis, with each Edge Shared by At Most 2 Cycles
- When does a biregular graph for the free product 2∗(2×2) have a 4 cycle?
- How does the divisibility graphs work?
- Euler's formula for tetrahedral mesh

And if a single HC is not possible, what is the smallest number and how does the corresponding graph look like?

- Twilight Zelda Guardian Puzzle : Shortest Path (UPDATE: ADDED RULES)
- Planar graphs with $n \geq 2$ vertices have at least two vertices whose degree is at most 5
- Is there a relationship between local prime gaps and cyclical graphs?
- Why this graph has automorphism group is isomorphic to the cyclic group of order 4?
- Proof related to breadth first search
- $K_6$ contains at least two monochromatic $K_3$ graphs.
- Round-robin party presents (or: Graeco-Latin square with additional cycle property)
- How many possible DAGs are there with $n$ vertices
- isomorphic planar graph with its complement
- Increase by one, Shortest path, changes the edges or not?

This paper here seems to suggest that this is impossible. Theorem 1 states that every cubic Hamiltonian graph has at least three Hamiltonian cycles and theorem 10 states that if the graphs are additionally bipartite then they have at least four Hamiltonian cycles.

The article contains quite a few other results which pertain to the subject and it might be of interest to you to peruse through.

Based on EuYu, it appears there is no such graph. I originally deleted my answer after that. But, now I put it back so the code is visible. This code will find examples of graphs that satisfy all the conditions EXCEPT the unique Hamilton cycle.

Using Sage, including the nauty generator of graphs, we can generate all graphs with 3 * ord / 2 edges easily. Of course, we only need check graphs with even order as well. So, this should go quick for graphs of small order. But, even at order 12 it might take a very long time. There are over 10 billion graphs of order 12, not sure how many have 18 edges exactly.

```
# ord must be even
ord = 12
num_edges = 3 * (ord/2)
# This is the string that gets put in the nauty generator to tell it which
# graphs to generate
# E.g., "12 18:18 -c" says generate order 12 graphs with 18 edges that are connected.
nauty_string = str(ord) + " " + str(num_edges) + ":" + str(num_edges) + " -c"
for g in graphs.nauty_geng(nauty_string):
# check min degree is 3. Since we have the constraint on edges above
# this should guarantee the max degree is 3 as well.
deg = g.degree()
deg.sort()
if deg[0] == 3:
if g.is_bipartite():
if g.is_planar():
vert_conn = g.vertex_connectivity()
if vert_conn >=2:
if vert_conn <=3:
if g.is_hamiltonian():
print g.graph6_string()
print "Finished with", ord
```

- Combinatorial argument for $\sum\limits_{k=i}^{n}\binom{n}{k}\binom{k}{i} = \binom{n}{i}2^{n-i}$
- Prove triangle inequality using the properties of absolute value
- Finding Jordan basis of a matrix ($3\times3$ example)
- Singular and Sheaf Cohomology
- Continuity of polynomials of two variables
- knowledge needed to understand Fermat's last theorem proof
- $\sum_{n=1}^{\infty}\frac{n^2}{a^2_{1}+a^2_{2}+\cdots+a^2_{n}}$is also convergent?
- What's the fastest way to tell if a function is uniformly continuous or not?
- To fast invert a real symmetric positive definite matrix that is almost similar to Toeplitz
- How to solve the Brioschi quintic in terms of elliptic functions?
- Beginner – Mathematical induction – help understanding example?
- Dominated convergence and $\sigma$-finiteness
- Summing $\frac{1}{e^{2\pi}-1} + \frac{2}{e^{4\pi}-1} + \frac{3}{e^{6\pi}-1} + \cdots \text{ad inf}$
- biased random walk on line
- The roots of $t^5+1$