On strong odd colorings of graphs

A strong odd coloring of a simple graph G is a proper coloring of the vertices of G such that for every vertex v and every color c, either c is used an odd number of times in the open neighborhood $N_G\left(v\right)$ or no neighbor of v is colored by c. The smallest integer k for which G admits a strong odd coloring with k colors is the strong odd chromatic number, ${\chi }_{\mathrm{so}}\left(G\right)$. These coloring notion and graph parameter were recently defined in Kwon and Park (arXiv:2401.11653). We answer a question raised by the originators concerning the existence of a constant bound for the strong odd chromatic number of all planar graphs. We also consider strong odd colorings of trees, unicyclic graphs, claw-free graphs, and graph products.