can find a strong 5-edge-coloring, improving Theorem \[thm:bi\].

\[thm:sparse\] Let $G$ be a subcubic graph.

1.  If $G$ does not contain $S_3$, $S_4$, or $S_7$ and $\operatorname{mad}(G) < 2 + \frac{1}{7}$, then ${\chi_{s}'(G)}\leq 5$.

2.  If $G$ is planar and has girth at least $30$, then ${\chi_{s}'(G)}\leq 5$.

The bound in Theorem \[thm:sparse\] is likely not sharp, but is close to optimal. The graph in Figure \[fig:thetaexample\] is subcubic, avoids $S_3$, $S_4$, and $S_7$, and satisfies both ${\chi_{s}'(G)}= 6$ and $\operatorname{mad}(G) = 2 + \frac{1}{6}$.

(0,0) node\[insep\](a) (5,0) node\[insep\](b)

\(a) to\[bend left=50\] node\[insep,pos=0.25\](x) node\[insep,pos=0.50\](y) node\[insep,pos=0.75\](z) (b) (x)– ++(90:0.5)node\[insep\] (y)– ++(90:0.5)node\[insep\] (z)– ++(90:0.5)node\[insep\]

\(a) to\[bend right=50\] node\[insep,pos=0.25\](x) node\[insep,pos=0.50\](y) node\[insep,pos=0.75\](z) (b) (x)– ++(-90:0.5)node\[insep\] (y)– ++(-90:0.5)node\[insep\] (z)– ++(-90:0.5)node\[insep\]

\(a) to node\[insep,pos=0.20\](x) node\[insep,pos=0.40\](y) node\[insep,pos=0.60\](z) node\[insep,pos=0.80\](w) (b) (x)– ++(90:0.5)node\[insep\] (y)– ++(90:0.5)node\[insep\] (z)– ++(90:0.5)node\[insep\] (w)– ++(90:0.5)node\[insep\] ;

;

Using similar methods, we improve the bounds in Theorem \[thm:wz\] when $d = 4$.

\[thm:sparse4\] Let $G$ be a subquartic graph.

1.  If $G$ has girth at least $7$ and $\operatorname{mad}(G) < 2 + \frac{2}{13}$, then $\chi_s'(G) \leq 7$.

2.  If $G$ is planar and has girth at least $28$, then $\chi_s'(G) \leq 7$.

We also consider a list variation of the strong chromatic index of $G$, first 