\[prism\], is a subcubic planar graph with ${\chi_{s}'(G)}= 9$.

(0,0)–(9,0) (0,0)–(0,4) (0,4)–(9,4) (9,4)–(9,0);

(0,0) node\[insep\] (0,4) node\[insep\] (9,4) node\[insep\] (9,0) node\[insep\] (2,2) node\[insep\] (7,2) node\[insep\];

(0,0)–(2,2) (0,4)–(2,2) (2,2)–(7,2) (7,2)–(9,4) (7,2)–(9,0);

Several papers prove sharper bounds on the strong chromatic index of planar graphs with additional structure [@Fouquet84; @Hocquard11; @Hocquard13; @Hudak], generally by introducing conditions on maximum average degree or girth to ensure that the target graph is sufficiently sparse. For a graph $G$, the [*maximum average degree*]{} of $G$, denoted $\operatorname{mad}(G)$, is the maximum of average degrees over all subgraphs of $G$. Hocquard, Montassier, Raspaud, and Valicov [@Hocquard11; @Hocquard13] proved the following.

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

1.  If $\operatorname{mad}(G) < \frac{7}{3}$, then ${\chi_{s}'(G)}\leq 6$. \[part1\]

2.  If $\operatorname{mad}(G) < \frac{5}{2}$, then ${\chi_{s}'(G)}\leq 7$. \[part2\]

3.  If $\operatorname{mad}(G) < \frac{8}{3}$, then ${\chi_{s}'(G)}\leq 8$.

\[thm:Hoc\]

(0,0)–(0,2) (0,2)–(2,2) (2,2)–(2,0) (2,0)–(0,0);

(0,0) node\[insep\] (0,2) node\[insep\] (2,2) node\[insep\] (2,0) node\[insep\] (1,2) node\[insep\] (1,3.5) node\[insep\] (0,2)–(1,3.5) (1,3.5)–(2,2);

\[fig:house\]

(-1.5,0)–(1.5,0) (-1.5,0) node\[insep\] (-.5,0) node\[insep\] (.5,0) node\[insep\] (1.5,0) node\[insep\] (90:1.5) node\[insep\] (270:1.5) node\[insep\] (-1.5,0)–(90:1.5) (90:1.5)–(1.5,0) (1.5,0)–(270:1.5) (270:1.5)–(-1.5,0) (90:1.5)–(2.5,1.5) (2.5,1.5)–(2.5,-1.5) (270:1.5)–(2.5,-1.5) (2.5,1.5) node\[insep\] (2.5,-1.5) node\[insep\] ;

\[fig:diamond\]

Parts (\[part1\]) and (\[part2\]) of Theorem \[thm:hocquard\] are sharp by the graphs shown in Figures \[fig:house\] and \[fig:diamond\], respectively. 