the minimum circulation is integral and will therefore still be zero.)

If $OPT_{LP}(G) =n$ and $G$ is a subquartic graph, then $G$ has a TSP tour of length at most $4n/3$.

Subquartic graphs: General case {#sec:gen}
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In this section, we consider the general case of subquartic graphs. For a graph $G=(V,E)$, [^2] suppose $OPT_{LP}(G) = (1+\eps)n$ for some $\eps > 0$. There is a fixed greedy DFS tree $T$ as defined in Section \[sec:stc\]. If we assign values to the edges in $B(T)$, then the only vertices that can add to the cost function are the expensive vertices, as we have defined them, since the maximum value allowed on an edge is one. Let $x(i,j) =
x_{ij}$ for all back edges in $B(T)$. Recall that the $\{x_{ij}\}$ values are obtained from the solution to $LP(G)$ in Section \[sec:LP\].

\[lem:tree-cut-sat\] A vertex $v$ in $T_{int}$ has at most one outgoing tree edge whose corresponding tree cut is not satisfied by $x$.

If $v$ has only one outgoing tree edge, then the lemma holds. Suppose $v$ is a branch vertex. First, let us consider the case when $v$ has three outgoing tree edges. In this case, if we consider the tree 