trivial upper bound of $2\Delta(G)(\Delta(G)-1)+1$ by showing that every graph $G$ with sufficiently large maximum degree satisfies ${\chi_{s}'(G)}\leq1.998\Delta(G)^2$. Bruhn and Joos [@BJ] have announced an improvement, claiming ${\chi_{s}'(G)}\leq 1.93\Delta(G)^2$.

The focus of this paper is the study of strong edge-colorings of *subcubic graphs*, those with maximum degree at most three, and *subquartic graphs*, those with maximum degree at most four. Faudree, Gyárfas, Schelp, and Tuza [@Faudree90] studied ${\chi_{s}'(G)}$ in the class of subcubic graphs, and gave the following conjectures.

Let $G$ be a subcubic graph.

(1) ${\chi_{s}'(G)}\leq 10$.

(2) If $G$ is bipartite, then ${\chi_{s}'(G)}\leq 9$. \[con:bipartite\]

(3) If $G$ is planar, then ${\chi_{s}'(G)}\leq 9$. \[con:planar\]

(4) If $G$ is bipartite and for each edge $xy \in E(G)$, $d(x) + d(y) \leq 5$, then ${\chi_{s}'(G)}\leq 6$.

(5) If $G$ is bipartite and $C_4 \not \subset G$, then ${\chi_{s}'(G)}\leq 7$.

(6) If $G$ is bipartite and its girth is large, then ${\chi_{s}'(G)}\leq 5$. \[con:girth\]

\[con:Faud\]

Several of these conjectures have been verified, including (1) by Andersen [@Andersen] and (2) by Steger and Yu [@StegerYu]. Quite recently, Kostochka, Li, Ruksasakchai, Santana, Wang, and Yu [@KLRSWY2014] announced an affirmative resolution to (3). This result is best possible since the prism, shown in Figure 