Rainbow vertex-connection number on a small-world Farey graph

Figures & data

Figure 1. A vertex coloring giving rainbow-connected $\mathcal{F}(2)$ and $\mathcal{F}(3).$

Figure 2. The ${\mathrm{c}}_{\mathrm{t}}$-coloring of S in $\mathcal{F}(t)$ where $t\ge 6.$

Table 1. List of (u₁, v₁)-path $P^{(0)}$ and its colors when $c_t(u_1)\ne c_t(v_1).$

u1	v1	$P^{(0)}$	colors of $P^{(0)}$
$\frac{0}{1}$	$\frac{3}{5}$	$\frac{0}{1},\frac{1}{2},\frac{3}{5}$	$a_1,a_0,a_2$
$\frac{0}{1}$	$\frac{3}{4}$	$\frac{0}{1},\frac{1}{1},\frac{3}{4}$	$a_1,a_0,a_2$
$\frac{1}{3}$	$\frac{1}{1}$	$\frac{1}{2},\frac{0}{1},\frac{1}{1}$	$a_2,a_1,a_0$
$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{3},\frac{1}{2},\frac{2}{3}$	$a_2,a_0,a_1$
$\frac{1}{2}$	$\frac{3}{4}$	$\frac{1}{2},\frac{2}{3},\frac{3}{4}$	$a_0,a_1,a_2$
$\frac{3}{5}$	$\frac{1}{1}$	$\frac{3}{5},\frac{2}{3},\frac{1}{1}$	$a_2,a_1,a_0$

Table 2. List of (u₁, v₁)-path $P^{(0)}$ and its colors when $c_t(u_1)=c_t(v_1).$

$c_t(u_1)$	$P^{(0)}$	colors of $P^{(0)}$
a1	$\frac{0}{1},\frac{1}{2},\frac{2}{3}$	$a_1,a_0,a_1$
a2	$\frac{1}{3},\frac{1}{2},\frac{3}{5}$	$a_2,a_0,a_2$
	$\frac{3}{5},\frac{2}{3},\frac{3}{4}$	$a_2,a_1,a_2$
	$\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}$	$a_2,a_0,a_1,a_2$