[@DBLP:conf/focs/MomkeS11], we can assume that these graphs are 2-vertex connected. The best-known approximation guarantee for these graphs is inherited from the general case, even when the graph is 4-regular, and is therefore $7/5$ due to §and Vygen. For subquartic graphs, we give an improved upper bound on the minimum cost of a circulation for $C(G,T)$. Using Lemma \[lem:MS\], this leads to an improved approximation guarantee of $46/33$ for graph-TSP on these graphs. Before we give an overview of our approach, we first explain our motivation for studying graph-TSP on this restricted class of graphs.

As mentioned in Section \[sec:prog\], graph-TSP is now known to be approximable to within $4/3$ for subcubic graphs. So, on the one hand, trying to prove the same guarantee for subquartic graphs is arguably a natural next step. Additionally, it is a well-motivated problem to study the graph-TSP on sparse graphs, because the support of an optimal extreme point solution to the standard linear programming relaxation (reviewed in Section \[sec:LP\]) has at most $2n-1$ non-zero edges (see Theorem 4.9 in [@cornuejols1985traveling]). Thus, any graph that corresponds to the support of such an an optimal solution to the standard linear program has average degree less than four.

