  After a sequence of improvements Boyd, Sitters, van der Ster, and Stougie
proved that any 2-connected graph whose n vertices have degree 3, i.e., a cubic
2-connected graph, has a Hamiltonian tour of length at most (4/3)n,
establishing in particular that the integrality gap of the subtour LP is at
most 4/3 for cubic 2-connected graphs and matching the conjectured value of the
famous 4/3 conjecture. In this paper we improve upon this result by designing
an algorithm that finds a tour of length (4/3 - 1/61236)n, implying that cubic
2-connected graphs are among the few interesting classes of graphs for which
the integrality gap of the subtour LP is strictly less than 4/3. With the
previous result, and by considering an even smaller epsilon, we show that the
integrality gap of the TSP relaxation is at most 4/3 - epsilon, even if the
graph is not 2-connected (i.e. for cubic connected graphs), implying that the
approximability threshold of the TSP in cubic graphs is strictly below 4/3.
Finally, using similar techniques we show, as an additional result, that every
Barnette graph admits a tour of length at most (4/3 - 1/18)n.
