Abstract:The Traveling Salesman Problem (TSP) is one of the most well-known NP-hard optimization problems. Following a recent trend of research which focuses on developing algorithms for special types of TSP instances, namely graphs of limited degree, in an attempt to reduce a part of the time and space complexity, we present a polynomialspace branching algorithm for the TSP in an n-vertex graph with degree at most 5, and show that it has a running time of O * (2.3500 n ), which improves the previous best known time bound of O * (2.4723 n ) given by the authors (the 12 th International Symposium on Operations Research and Its Application (ISORA 2015(ISORA ), pp.45-58, 2015. While the base of the exponent in the running time bound of our algorithm is greater than 2, it still outperforms Gurevich and Shelah's O * (4 n n log n ) polynomial-space exact algorithm for the TSP in general graphs (SIAM Journal of Computation, Vol.16, No.3, pp.486-502, 1987). In the analysis of the running time, we use the measure-and-conquer method, and we develop a set of branching rules which foster the analysis of the running time.