However, our actual motivation for studying graphs with degree at most four has more to do with understanding the Mömke-Svensson algorithm than with an abstract interest in subquartic graphs. The basic approach to computing an upper bound on the minimum cost circulation in $C(G,T)$ used in both [@DBLP:conf/focs/MomkeS11] and [@DBLP:conf/stacs/Mucha12] is to specify flow values on the edges of $C(G,T)$ that are functions of an optimal solution to the linear programming relaxation for graph-TSP on the graph $G$. The cost of the circulation obtained using these values can be analyzed in a local, vertex by vertex manner. Mucha showed that vertices with degree four or five potentially increase the cost of the circulation the most [@DBLP:conf/stacs/Mucha12]. In fact, one could hypothetically construct a tight example for Mucha’s analysis of the Mömke-Svensson algorithm on a graph where each vertex has degree at most four (or where each vertex has degree at most five). It therefore seems worthwhile to determine if the cost of the circulation can be improved on subquartic graphs. Our results actually hold for a slightly more general class of graphs than subquartic graphs: they hold for any graph that has an optimal solution to the standard linear programming 