A proof of a version of Griggs and Yeh’s conjecture for L(2,1)-labeling of iterated Mycielskians

In a graph $G=(V\left(G\right),E\left(G\right))$, a function $f$ from the vertex set $V\left(G\right)$ to the set of all nonnegative integers is called an $L(2,1)$-labeling of $G$, if it satisfies the following conditions for any two vertices $x$ and $y$, $|f\left(x\right)-f\left(y\right)|\ge 2$ when $d_G(x,y)=1$, and $|f\left(x\right)-f\left(y\right)|\ge 1$ when $d_G(x,y)=2$, where $d_G(x,y)$ denotes the distance between the vertices $x$ and $y$ in $G$. The span of $f$ is the difference between the largest and the smallest label used by $f$. The $L(2,1)$-labeling number of $G$ is the minimum span over all $L(2,1)$-labelings of $G$, it is denoted by ${\lambda }_{2,1}\left(G\right)$. In 1992, Griggs and Yeh conjectured that for any graph $G$ of maximum degree $\Delta \left(G\right)\ge 2$, the inequality ${\lambda }_{2,1}\left(G\right)\le \Delta {\left(G\right)}^2$ holds. This conjecture has been extensively studied over the past decades and solved in numerous specific cases, yet it remains unsolved in general. In Dliou et al. (2024), the authors studied the $L(2,1)$-labeling number of the iterated Mycielskian of graphs. The statement of Griggs and Yeh’s conjecture led them to conjecture that for any $t\ge 1$, we have ${\lambda }_{2,1}\left(M^t\left(G\right)\right)\le (2^t-1)(\left|G\right|+1)+\Delta {\left(G\right)}^2$, where $M^t\left(G\right)$ denotes the $t$th iterated Mycielskian of a simple graph $G$. In this note, we prove this conjecture for all $t\ge 2$. Moreover, we show that for $t\ge 2$, the upper bound $(2^t-1)(\left|G\right|+1)+\Delta {\left(G\right)}^2$ is achieved only when $G$ is a single edge or a diameter 2 Moore graph, which exist only when $\Delta \left(G\right)=2,3,7,$ and perhaps for $\Delta \left(G\right)=57$.