An identifying operation on a 1-planar graph with an application to acyclic coloring

This paper introduces a graph operation and gives its applications. Given a 1-plane graph M and its crossing point x formed by two crossing edges $u{u}^{\prime }$ and $v{v}^{\prime }$, an Identifying Operation with respect to x is defined in two steps: (1) identifying u and v such that x vanishes; (2) deleting loops and multi-edges (if exists). Using Identifying Operation to every crossing point, we change M into its associated plane graph $M^{⁎}$. Under some conditions, we show that ${\chi }_a\left(M\right)\le 2{\chi }_a\left(M^{⁎}\right)$, where the parameters ${\chi }_a\left(M\right)$ and ${\chi }_a\left(M^{⁎}\right)$ represent the acyclic-chromatic-number of M and $M^{⁎}$, respectively. This generalizes a result established by Yang et al. in 2018.