show how to sample an odd-join of size $|E'|/3$, where $E'$ is the subset of edges contained in the circulation network $C(G,T)$. The number of edges that are doubled to guarantee connectivity is directly related to the minimum cost of a circulation of $C(G,T)$. Lemma 4.1 from [@DBLP:conf/focs/MomkeS11] relates this cost to the size of a solution output by their algorithm:

[**[[@DBLP:conf/focs/MomkeS11]]{}**]{}\[lem:MS\] Given a 2-vertex connected graph $G$ and a depth first search tree $T$ of $G$, let $C^*$ be a minimum cost circulation for $C(G,T)$ of cost $c(C^*)$. Then there is a spanning Eulerian multigraph in $G$ with at most $\frac{4}{3}n + \frac{2}{3} c(C^*)$ edges.

We defer a precise description of the circulation network $C(G,T)$ to Section \[sec:prelim\], where we formulate it using different notation from that in [@DBLP:conf/focs/MomkeS11]. For the moment, we emphasize that if one can prove a better upper bound on the value of $c(C^*)$, then this directly implies an improved upper bound on the number of edges in a tour output by Mömke and Svensson’s algorithm.

Our contribution
----------------

We consider the graph-TSP problem for [*subquartic*]{} graphs, i.e. graphs in which each vertex has degree at most four. As pointed out in Lemma 2.1 of 