Previously, the authors of this work have presented in a series of
papers polynomial-space algorithms for the TSP in graphs with degree
at most five, six and seven. Each of these algorithms is the first
algorithm specialized for the TSP in graphs of limited degree five,
six, and seven respectively, and the running time bound of these
algorithms outperforms Gurevich and Shelah's $O^*(4^n n^{\log n})$
algorithm for the TSP in $n$-vertex graphs (SIAM Journal of
Computation, 16(3), pp. 486--502, 1987). Now we ask what is the
highest degree~$i$ until which a specialized polynomial-space
algorithm for the TSP in graphs with maximum degree~$i$ outperforms
Gurevich and Shelah's $O^*(4^n n^{\log n})$ algorithm?
As an answer to this question, this paper presents the first
polynomial-space exact algorithm specialized for the TSP in graphs
with degree at most eight. We develop a set of branching rules to aid
the analysis of the branching algorithm, and we use the
measure-and-conquer method to effectively analyze our branching
algorithm. We obtain a running time of $O^*(4.1485^n)$, and this
running time bound does not give an advantageous algorithm for the TSP
in degree-8 graphs over Gurevich and Shelah's algorithm for the TSP in
general, but it gives a limit as to the applicability of our choice of
branching rules and analysis method for designing a polynomial-space
exact algorithm for the TSP in graphs of limited degree.