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

TL;DR

This work advances understanding of $L(3,2,1)$-labelings for three families of 4-valent circulants $C_n(\{1,s_2,n-s_2,n-1\})$ with $s_2\in\{3,4,5\}$. It combines lower bounds, constructive pattern labelings, and substantial computer-aided searches to determine many exact labeling numbers $\lambda_{(3,2,1)}$ and to propose general conjectures tied to modular conditions on $n$ (e.g., divisibility by 12, 16). The authors employ Sylvester-type decomposition and concatenation of short labeling patterns to build feasible labelings, supported by extensive tables and patterns. The results yield tight bounds and several conjectures that guide future theoretical analysis in the area of circulant $L(3,2,1)$-labelings with practical implications for frequency/channel assignments.

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 $λ_{(3,2,1)}(G)$. In this paper, we study $L(3,2,1)$-labelings of three classes of circulant graphs. Namely, we investigate $λ_{(3,2,1)}$ of $C_n(\{1,s_2,n-s_2,n-1\})$, where $s_2\in\{3,4,5\}$. This paper is a continuation of a recent publication of T. Calamoneri who studied the square of cycles, i.e., circulants $C_n(\{1,2,n-2,n-1\})$.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Proposition 2.1
• proof
• Theorem 2.2