Incidence coloring of outer-1-planar graphs (Q6565538)

A proper incidence $k$-coloring of a graph $G$ is a coloring of the incidences using $k$ colors in such a way that every two adjacent incidences have distinct colors. The minimum integer $k$ such that $G$ has a proper incidence $k$-coloring is the incidence chromatic number of $G$, denoted by $\chi_{i}(G)$. An incidence $(k,l)$-coloring of $G$ is a proper incidence $k$-coloring such that $|A_v|\leq l$ for each $v\in V(G)$. The authors provide the following conjecture. \N\NConjecture 1. $\chi_{i}(G) \leq \Delta(G)+2 $ holds for every planar graph $G$.\N\NThe authors confirm the conjecture for outer-1-planar graphs $G$ with $\Delta(G) \geq 8$ or $g(G) \geq 4$. Specifically, they prove the following results.\N\NTheorem 1. Every outer-1-planar graph $G$ has an incidence $(\Delta(G) + 3, 2)$-coloring.\N\NTheorem 2. Every outer-1-planar graph $G$ with $\Delta(G)\ge 8$ has an incidence $(\Delta(G) + 2, 2)$- coloring. \N\NTheorem 3. Every outer-1-planar graph $G$ with $g(G)\ge 4$ has an incidence $(\Delta(G) + 2, 2)$- coloring.