Summary: The Travelling Salesman Problem is one of the fundamental and intensively studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides’s algorithm with an approximation factor of \(\frac{3}{2}\), even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality gap of only \(\frac{4}{3}\). Very recently, significant progress has been made for the important special case of graphic metrics, first by S. Oveis Gharan et al. [in: Proceedings of the 2011 IEEE 52nd annual symposium on foundations of computer science, FOCS 2011. Los Alamitos, CA: IEEE Computer Society. 550–559 (2011; Zbl 1292.68171)], and then by T. Mömke and O. Svensson [ibid. 560–569 (2011; Zbl 1292.68169)]. In this paper, we provide an improved analysis of the approach presented by Mömke and Svensson [loc. cit.] yielding a bound of \(\frac{13}{9}\) on the approximation factor, as well as a bound of \(\frac{19}{12}+\varepsilon\) for any \(\varepsilon>0\) for a more general Travelling Salesman Path Problem in graphic metrics.