Semistrong edge colorings of planar graphs

Strengthened notions of a matching M of a graph G have been considered, requiring that the matching M has some properties with respect to the subgraph $G_M$$G_M$ of G induced by the vertices covered by M: If M is the unique perfect matching of $G_M,$$G_M,$ then M is a uniquely restricted matching of G; if all the edges of M are pendant edges of $G_M,$$G_M,$ then M is a semistrong matching of G; if all the vertices of $G_M$$G_M$ are pendant, then M is an induced matching of G. Strengthened notions of edge coloring and of the chromatic index follow.

In this paper, we consider the maximum semistrong chromatic index of planar graphs with given maximum degree $\mathrm{\Delta }.$$\Delta .$ We prove that graphs with maximum average degree less than 14/5 have semistrong chromatic index (hence uniquely restricted chromatic index) at most $2\mathrm{\Delta }+4,$$2\Delta +4,$ and we reduce the bound to $2\mathrm{\Delta }+2$$2\Delta +2$ if the maximum average degree is less than 8/3. These cases cover, in particular, the cases of planar graphs with girth at least 7 (resp. at least 8).

Our result makes some progress on the conjecture of Lužar et al. (J Graph Theory 105:612–632, 2024), which asserts that every planar graph G has a semistrong edge coloring with $2\mathrm{\Delta }+C$$2\Delta +C$ colors, for some universal constant C. (Note that such a conjecture would fail for strong edge coloring as there exist graphs with arbitrarily large maximum degree that are not strongly $(4\mathrm{\Delta }-5)$$(4\Delta -5)$-edge-colorable.) We provide an example of a planar graph showing that the maximum semistrong chromatic index of planar graphs with maximum degree $\mathrm{\Delta }$$\Delta$ is at least $2\mathrm{\Delta }+4.$$2\Delta +4.$