Are all $4$-regular graphs Hamiltonian It is easy to show that all connected $2$-regular graphs are Hamiltonian. The Petersen graph is a $3$-regular graph that is not Hamiltonian. 
Are there any $4$ regular graphs that are not Hamiltonian?
Thank you
 A: Note that you didn't specify connected graph, which is certainly necessary for Hamiltonicity.  More generally, a Hamilton cycle has the property that deleting any $k$ vertices breaks the cycle into at most $k$ connected components.  In the case $k=1$ a Hamiltonian graph must be strongly connected.
Pick your favourite $4$-regular graph $G$, and delete one edge $e$ so it has two vertices of degree $3$.  Take two copies of $G-e$, and create a new vertex that is adjacent to the four deficient vertices.  The resulting graph is $4$-regular, connected, but not strongly connected.
Also, did you try Googling 4-regular non-Hamiltonian?  The Meredith graph is very strongly connected, $4$-regular and still not Hamiltonian.
A: However, Tutte proved that a 4-connected planar graph is Hamiltonian.
A: The answer is no. The graph on the picture is the smallest counterexample
You can most likely generalize this construction for every $k > 4$ and deduce that you can always find connected $k$-regular graphs that are not Hamiltonian.
