$L(2,1)$-Labeling of the iterated Mycielski of graphs and some related to matching problems

TL;DR

This paper investigates the $L(2, 1)$-labeling of the Mycielski and iterated Mycielski graphs, providing bounds, characterizations, and specific results for various graph classes, advancing understanding of graph labeling complexities.

Contribution

It establishes sharp bounds for the $L(2, 1)$-labeling number of iterated Mycielski graphs and characterizes graphs achieving these bounds, including conditions related to matching numbers.

Findings

Bounds for $(M^t(G))$ in terms of $t$, $n$, and $(G)$

Characterization of graphs achieving upper and lower bounds

Determination of $$ for specific graph classes

Abstract

In this paper, we study the $L(2, 1)$-Labeling of the Mycielski and the iterated Mycielski of graphs in general. For a graph $G$ and all $t\geq 1$, we give sharp bounds for $\lambda(M^t(G))$ the $L(2, 1)$-labeling number of the $t$-th iterated Mycielski in terms of the number of iterations $t$, the order $n$, the maximum degree $\bigtriangleup$, and $\lambda(G)$ the $L(2, 1)$-labeling number of $G$. For $t=1$, we present necessary and sufficient conditions between the $4$-star matching number of the complement graph and $\lambda(M(G))$ the $L(2, 1)$-labeling number of the Mycielski of a graph, with some applications to special graphs. For all $t\geq 2$, we prove that for any graph $G$ of order $n$, we have $2^{t-1}(n+2)-2\leq \lambda(M^t(G))\leq 2^{t}(n+1)-2$. Thereafter, we characterize the graphs achieving the upper bound $2^t(n+1)-2$, then by using the Marriage Theorem and Tutte's characterization of graphs with a perfect $2$-matching, we characterize all graphs without isolated vertices achieving the lower bound $2^{t-1}(n+2)-2$. We determine the $L(2, 1)$-labeling number for the Mycielski and the iterated Mycielski of some graph classes.