Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Radio Labeling of Strong Prismatic Network With Star

Liming Wang, Feng Li, Linlin Cui

TL;DR

This paper addresses the radio labeling problem on the strong prismatic network with star $G = S_n \boxtimes C_m$, a graph model relevant to spectrum assignment in wireless networks. It derives tight, parity-dependent bounds and exact values for the radio number $rn(G)$ by decomposing $G$ into substars and analyzing critical paths, producing explicit labelings that achieve these bounds. The key results are $rn(G)=\dfrac{m^{2} + m(m-1) + (n-1)(m-2)}{2}$ for even $m$ and $rn(G)=\dfrac{m^{2} + 4mn + 5m - 8n + 18}{2}$ for odd $m$, with constructive proofs and optimal labeling schemes. A parallel algorithm is proposed to compute labelings efficiently for large-scale networks, supported by visualization that demonstrates the dependence on $n$ and $m$ and the impact of parity. These findings provide a solvable case for a challenging NP-hard problem and offer practical guidance for efficient channel assignment in regular, high-diameter network topologies.

Abstract

The rapid development of wireless communication has made efficient spectrum assignment a crucial factor in enhancing network performance. As a combinatorial optimization model for channel assignment, the radio labeling is recognized as an NP-hard problem. Therefore, converting the spectrum assignment problem into the radio labeling of graphs and studying the radio labeling of specific graph classes is of great significance. For $G$, a radio labeling $\varphi: V(G) \to \{0, 1, 2, \ldots\}$ is required to satisfy $|\varphi(u) - \varphi(v)| \geq \text{diam}(G) + 1 -d_G(u, v)$, where ${diam(G)}$ and $d_G(u, v)$ are diameter and distance between $u$ and $v$. For a radio labeling $\varphi$, its $\text{span}$ is defined as the largest integer assigned by $\varphi$ to the vertices of $G$; the radio labeling specifically denotes the labeling with the minimal span among possible radio labeling. The strong product is a crucial tool for constructing regular networks, and studying its radio labeling is necessary for the design of optimal channel assignment in wireless networks. Within this manuscript, we discuss the radio labeling of strong prismatic network with star, present the relevant theorems and examples, and propose a parallel algorithm to improve computational efficiency in large-scale network scenarios.

Radio Labeling of Strong Prismatic Network With Star

TL;DR

This paper addresses the radio labeling problem on the strong prismatic network with star , a graph model relevant to spectrum assignment in wireless networks. It derives tight, parity-dependent bounds and exact values for the radio number by decomposing into substars and analyzing critical paths, producing explicit labelings that achieve these bounds. The key results are for even and for odd , with constructive proofs and optimal labeling schemes. A parallel algorithm is proposed to compute labelings efficiently for large-scale networks, supported by visualization that demonstrates the dependence on and and the impact of parity. These findings provide a solvable case for a challenging NP-hard problem and offer practical guidance for efficient channel assignment in regular, high-diameter network topologies.

Abstract

The rapid development of wireless communication has made efficient spectrum assignment a crucial factor in enhancing network performance. As a combinatorial optimization model for channel assignment, the radio labeling is recognized as an NP-hard problem. Therefore, converting the spectrum assignment problem into the radio labeling of graphs and studying the radio labeling of specific graph classes is of great significance. For , a radio labeling is required to satisfy , where and are diameter and distance between and . For a radio labeling , its is defined as the largest integer assigned by to the vertices of ; the radio labeling specifically denotes the labeling with the minimal span among possible radio labeling. The strong product is a crucial tool for constructing regular networks, and studying its radio labeling is necessary for the design of optimal channel assignment in wireless networks. Within this manuscript, we discuss the radio labeling of strong prismatic network with star, present the relevant theorems and examples, and propose a parallel algorithm to improve computational efficiency in large-scale network scenarios.
Paper Structure (7 sections, 10 theorems, 76 equations, 8 figures, 1 table)

This paper contains 7 sections, 10 theorems, 76 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main Results
  4. Bounds for the radio labeling of the strong prismatic network with star contain even cycle
  5. Bounds for the radio labeling of the strong prismatic network with star contain odd cycle
  6. Optimal radio labeling of the strong prismatic network with star
  7. Conclusion

Key Result

Lemma 1.4

Let $G_1$ and $G_2$ be two connected undirected graph with $G=G_1\boxtimes G_2$ their strong product, then for any two separate vertices $u=(u_1,v_1)$, $v=(u_2,v_2)\in V(G)$, their distance in $G$ is: where $d_{G_1}(u_1, u_2)$ denotes the distance between $u_1$ and $u_2$ in $G_1$, and $d_{G_2}(v_1, v_2)$ denotes the distance between $v_1$ and $v_2$ in $G_2$.

Figures (8)

  • Figure 1: Strong prismatic network with 4-order stars.
  • Figure 2: Substar decomposition.
  • Figure 3: Substar decomposition.
  • Figure 4: Radio labeling of strong prismatic network with star when m = 6.
  • Figure 5: Radio labeling of strong prismatic network with star when m = 5.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 1.4: 2
  • Lemma 1.5: 2
  • Definition 1.6: 8
  • Lemma 2.1: 13
  • Lemma 2.2: 22
  • Definition 2.3
  • Definition 2.4
  • ...and 15 more