$L(1,1)-$ Labeling of Direct Product of Cycles

TL;DR

This paper determines the minimum label range for $L(1,1)$-labeling of the direct product of cycles, expanding understanding of graph labelings for various cycle combinations.

Contribution

It establishes the $mbda_1^1$-numbers for the direct product of cycles $C_m imes C_n$ for all positive integers $m,n \u2265 3$, covering cases where both are even or mixed parity.

Findings

Calculated $mbda_1^1$-numbers for even-even cycle products.

Determined $mbda_1^1$-numbers for mixed parity cycle products.

Extended known results to all positive cycle pairs with $m,n \u2265 3$.

Abstract

An $L(1,1)$-labeling of a graph $G$ is an assignment of labels from $\{0,1 \cdots, k \}$ to the vertices of $G$ such that two vertices that are adjacent or have a common neighbor receive distinct labels. The $\lambda_1^1-$ number, $\lambda_1^1(G)$ of $G$ is the minimum value $k$ such that $G$ admits an $L(1,1)$ labeling. We establish the $\lambda_1^1-$ numbers for direct product of cycles $C_m \times C_n$ for all positive $m, n \geq 3$, where both $m,n$ are even or when one of them is even and the other odd.