I would like to see how to show this statement is true. Can someone demonstrate why this is not Hamiltonian?
2026-03-25 04:43:22.1774413802
show that if $G$ is a $2$-connected graph containing a vertex that is adjacent to at least three vertices of degree $2$, then $G$ is not Hamiltonian.
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This is the obstruction to having a Hamilton cycle:
(The rest of the graph is not shown, but there are no additional edges with a red vertex as an endpoint.)
If the graph has a Hamilton cycle, then it does not use all three edges $xa$, $xb$, and $xc$ (or else it's not a cycle). Thus, we can delete the edge not used in the Hamilton cycle, and we still have a Hamiltonian graph.
However, after deleting any one of these three edges, one of the vertices $a$, $b$, or $c$ becomes a degree-$1$ vertex. This contradicts the graph being Hamiltonian (since all vertices in a Hamilton cycle have degree $2>1$).