r-dynamic colorings and the spectral radius in graphs

Let $\chi \left(G\right)$ and $\rho \left(G\right)$ be the chromatic number and spectral radius of $G$, respectively. In 1967, Wilf proved that for a graph $G$, we have $\chi \left(G\right)\le 1+\rho \left(G\right)$. An $r$-dynamic $k$-coloring of a graph $G$ is a proper $k$-coloring of $G$ such that every vertex $v$ in $V\left(G\right)$ has neighbors in at least $min\{d\left(v\right),r\}$ different color classes. The $r$-dynamic chromatic number of a graph $G$, written ${\chi }_r\left(G\right)$, is the least $k$ such that $G$ has such a $k$-coloring. Note that $\chi \left(G\right)={\chi }_1\left(G\right)$ and ${\chi }_r\left(G\right)\le 1+r\Delta \left(G\right)$ (*) (Jahanbekama et al., 2016). By the inequality (*), we observe that for a positive integer $r\ge 2$ and a connected graph $G$, we have ${\chi }_r\left(G\right)\le 1+r{\rho }^2\left(G\right).$

In this paper, for a positive integer $k>r^2$, we provide graphs $H_{k,r}$ with ${\chi }_r\left(H_{k,r}\right)=\Theta \left({\rho }^2\left(H_{k,r}\right)\right)$ to show that the bound is almost sharp. When $r=2$, we prove that ${\chi }_r\left(G\right)\le 1+{\rho }^2\left(G\right)$; equality holds only when $G=P_1,P_2,P_3,$ or $C_5$. For $r=3$ and $\Delta \left(G\right)\le 4$, we prove that ${\chi }_r\left(G\right)\le 10$; equality holds when $G$ is the Petersen graph. When $r=3$ and $\Delta \left(G\right)\ge 5$, we prove that ${\chi }_r\left(G\right)\le 2\Delta \left(G\right)+1$, which implies ${\chi }_r\left(G\right)\le 1+2{\rho }^2\left(G\right)$. The graph $H_{k,3}$ guarantees that ${\chi }_3\left(G\right)\le 2\Delta \left(G\right)+1$ is sharp.