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<p>Connectivity is a central concept in graph theory, often used to model relationships and communication in networks. A recent and intriguing variation is <i>rainbow connectivity</i>. It involves assigning colors to the edges of a graph so that every pair of vertices is connected by a path in which all edges have distinct colors, known as a rainbow path. This concept has applications in secure communication networks, where messages must travel across diverse, non-repeating routes. In this work, we examine the rainbow connectivity of two important families of graphs: generalized cycle graphs and wheel graphs. These structures are commonly found in both theoretical graph studies and real-world networks. We determine the optimal edge-colorings that ensure rainbow connectivity in these graphs and establish their corresponding rainbow connection numbers. Our results provide insight into how structural properties of graphs influence their connectivity under rainbow constraints, contributing to the broader study of graph coloring and network design.</p>

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Rainbow Coloring of Cycle of Generalized Cycle and Wheel Graphs for Secured Communication Networks

  • 期刊论文

  • 发表日期：

    2026年4月25日

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  • Madhu Nagamangala Rajendra,
  • Srinivasa Rao Karanalu,
  • Thirumalesh Kodandaramaiah,
  • Sangeetha Basavaraju,
  • Sunil Shreedhara Murthy

摘要

Connectivity is a central concept in graph theory, often used to model relationships and communication in networks. A recent and intriguing variation is rainbow connectivity. It involves assigning colors to the edges of a graph so that every pair of vertices is connected by a path in which all edges have distinct colors, known as a rainbow path. This concept has applications in secure communication networks, where messages must travel across diverse, non-repeating routes. In this work, we examine the rainbow connectivity of two important families of graphs: generalized cycle graphs and wheel graphs. These structures are commonly found in both theoretical graph studies and real-world networks. We determine the optimal edge-colorings that ensure rainbow connectivity in these graphs and establish their corresponding rainbow connection numbers. Our results provide insight into how structural properties of graphs influence their connectivity under rainbow constraints, contributing to the broader study of graph coloring and network design.