Moemke and Svensson presented a beautiful new approach for the traveling salesman problemon a graph metric (graph-TSP), which yields a 4/3-approximation guarantee on subcubic graphsas well as a substantial improvement over the 3/2-approximation guarantee of Christofides’algorithm on general graphs. The crux of their approach is to compute an upper bound on theminimum cost of a circulation in a particular network,C(G, T), whereGis the input graphandTis a carefully chosen spanning tree. The cost of this circulation is directly related tothe number of edges in a tour output by their algorithm. Mucha subsequently improved theanalysis of the circulation cost, proving that Moemke and Svensson’s algorithm for graph-TSPhas an approximation ratio of at most 13/9 on general graphs. This analysis of the circulation is local, and vertices with degree four or five can contributethe most to its cost. Thus, hypothetically, there could exist a subquartic graph (a graph withdegree at most four at each vertex) for which Mucha’s analysis of the M ̈omke-Svensson algorithmis tight. We show that this is not the case and that M ̈omke and Svensson’s algorithm for graph-TSP has an approximation guarantee of at most 25/18 on subquartic graphs. To prove this,we present different methods to upper bound the minimum cost of a circulation on the networkC(G, T). Our approximation guarantee holds for all graphs that have an optimal solution for astandard linear programming relaxation of graph-TSP with subquartic support.