L(3,2,1)-labelings of three classes of 4-valent circulants

TL;DR

This paper investigates the minimum span of $L(3,2,1)$-labelings for specific classes of 4-valent circulant graphs, extending previous work on the square of cycles.

Contribution

It provides new results on $L(3,2,1)$-labelings for circulant graphs $C_n(1,t)$ with $t=3,4,5$, expanding the understanding of labelings in these graph classes.

Findings

Determined $ ext{lambda}_{(3,2,1)}$ for $C_n(1,3)$, $C_n(1,4)$, and $C_n(1,5)$.

Extended previous results from $C_n(1,2)$ to other circulant graphs.

Contributed to the theory of graph labelings for 4-valent circulants.

Abstract

An $L(3,2,1)$-labeling of a graph $G$ is an assignment $f$ of nonnegative integers to vertices such that $\vert f(x)-f(y)\vert > 3-\mbox{dist}_G(x,y)$ for every pair $x,y$ of vertices of $G$, where $\mbox{dist}_G(x,y)$ denotes the distance between $x$ and $y$ in $G$. The minimum span (i.e., the difference between the largest and the smallest value) among all $L(3,2,1)$-labelings of $G$ is denoted by $\lambda_{(3,2,1)}(G)$. In this paper, we study $L(3,2,1)$-labelings of three classes of circulant graphs. Namely, we investigate $\lambda_{(3,2,1)}$ of circulant graphs $C_n(1,t)$, where $t\in\{3,4,5\}$ and $n$ is the order of the graph. This paper is a continuation of a recent publications of V. Bianco and T. Calamoneri who studied the square of cycles, i.e., circulant graphs $C_n(1,2)$.