Independence, matching and packing coloring of the iterated Mycielskian of graphs

Abstract

Let $\alpha \left(G\right)$, $\nu \left(G\right)$, ${\nu }_2\left(G\right)$ and ${\chi }_{\rho }\left(G\right)$ denote respectively the independence, matching, 2-matching and packing chromatic numbers of a graph $G$. A well-known construction on graphs, called the Mycielskian of a graph, transforms any $k$-chromatic graph $G$ into a $(k+1)$-chromatic graph $M\left(G\right)$ having an equal clique number to $G$. The $t$th iterated Mycielskian of a graph $G$, denoted $M^t\left(G\right)$, is obtained by iteratively repeating the Mycielskian transformation $t$ times. In this paper, we give $\alpha \left(M\left(G\right)\right)$ in terms of ${\nu }_2\left(G\right)$. Then we show that for all $t\ge 2$, $\alpha \left(M^t\left(G\right)\right)=max\{\left|M^{t-1}\left(G\right)\right|,2^{t-1}\alpha \left(M\left(G\right)\right)\}$. We characterize for all $t\ge 1$, the connected graphs having $\alpha \left(M^t\left(G\right)\right)=\left|M^{t-1}\left(G\right)\right|$ and those having $\alpha \left(M^t\left(G\right)\right)=2^t\alpha \left(G\right)$. Afterwards, we give $\nu \left(M^t\left(G\right)\right)$ and ${\nu }_2\left(M^t\left(G\right)\right)$ for all $t\ge 1$ in terms of ${\nu }_2\left(G\right)$. Then we show that for all $t\ge 1$, $M^t\left(G\right)$ is a König–Egerváry graph if and only if $G$ does not have a perfect 2-matching. Later, we investigate the packing chromatic number of $M^t\left(G\right)$. We present several sharp upper and lower bounds for ${\chi }_{\rho }\left(M^t\left(G\right)\right)$, some in terms of the number of iterations $t$, the order of $G$, the $k$-independence number with $k\in \{1,2,3\}$ and ${\chi }_{\rho }\left(G\right)$. We show that ${\chi }_{\rho }\left(M^t\left(G\right)\right)$ can be computed in polynomial time if $G$ has a diameter at most 2. Recently, in Bidine et al. (2023) the authors studied diameter two graphs having ${\chi }_{\rho }\left(M^t\left(G\right)\right)=2^t{\chi }_{\rho }\left(G\right)$ for $t\ge 1$. Here we fully characterize the diameter two graphs for which this equality holds. They also asked a question about the growth of ${\chi }_{\rho }\left(M^t\left(G\right)\right)$ in terms of ${\chi }_{\rho }\left(G\right)$. We show that for $t\ge 1$, ${\chi }_{\rho }\left(M^t\left(G\right)\right)$ cannot be upper bounded by a function of ${\chi }_{\rho }\left(G\right)$ alone. In addition, we discuss the realizable values for ${\chi }_{\rho }\left(M^t\left(G\right)\right)$ and characterize the graphs having the least possible ${\chi }_{\rho }\left(M^t\left(G\right)\right)$.