Hello, I am trying to prove that this graph does not have a Hamilton circuit. The only thing I know is that all vertices of degree 2 must have all their edges in the circuit, however this does not prove to be useful. Are there any other useful observations that can be made?
Proving a graph does not have a hamilton circuit
graph theoryhamiltonicity
Best Answer
There's two approaches here.
The first is to start from where you started from: the degree-$2$ vertices ($T$ and $P$) must all have their edges in the Hamiltonian cycle. Once you draw those edges, you can keep going with some other vertices. For example, you'll see that vertex $Y$ (which has $3$ edges) can no longer use one of them, so it must use the other two. Repeat until you paint yourself into a corner.
The second is a problem of global structure. The graph is bipartite. Identify how, and then figure out why this prevents a Hamiltonian cycle from existing.