## 1 Introduction  
One of the most stimulating areas of research today across science, engineering, and particularly mathematics involves identifying and studying novel connections between disciplines. Such interdisciplinary approaches enable researchers to address open problems, refine existing theories, and uncover new insights. This paper explores the intersection of graph theory and sensor networks. Sensor networks, comprising distributed systems of interconnected devices that collect and transmit data, represent a dynamic technological domain. Graph theory, in turn, provides a powerful framework for modeling and analyzing these complex systems. By leveraging graph theory, researchers have achieved significant advancements in understanding, optimizing, and managing sensor networks. In the context of the modern interconnected landscape, sensor networks play a crucial role pivotal in real-time data collection and transmission. Modeling these systems as graphs creates a synergy that facilitates efficient analysis, optimization, and management of network connectivity and information flow. Graph theory has already proven its utility in various fields, including sensor networks. For example, in [1] the authors study several graph-based optimization approaches such as node placement, routing protocols, and data aggregation to minimize energy consumption and extend network lifespan. Other studies employ graph algorithms for topology construction, energy-efficient communication, and operations optimization in wireless sensor networks [2]. Similarly, graph-based methods address challenges such as load balancing and connectivity [3; 4; 5]. This paper explores three fundamental concepts in graph theory: vertex coloring, Eulerian paths, and Hamiltonian paths and their practical applications in sensor networks. These tools have proven utility in tasks such as connectivity analysis, route optimization, task scheduling, resource management, and data collection. For instance, Eulerian cycles have been applied to optimize transportation routes [6], Hamiltonian paths have been used to design energy-efficient mobile charging mechanisms [7], and vertex coloring has been utilized to assign roles or tasks within sensor networks [8]. The main objective of this study is to advance the understanding and application of graph-based methods for improving the efficiency and robustness of sensor network management. Specifically, we aim to associate sensor networks with Delaunay graphs to analyze their structural and functional properties through three key graph tools: Eulerian trails, Hamiltonian paths, and chromatic numbers. While Hamiltonian paths in Delaunay graphs have been studied extensively under specific conditions, such as the \(L^{\infty}\)-metric [9], this work tackles several open questions related to Delaunay graphs. One key focus is the Eulerian nature of these graphs, investigating whether they can be classified as Eulerian or semi-Eulerian and under what specific conditions this classification holds. Another aspect explored is the chromatic number of Delaunay graphs, aiming to determine its bounds or exact values. Finally, this study examines the practical implications of these theoretical findings, particularly in enhancing the connectivity, efficiency, and energy management of sensor networks in real-world applications. To address these objectives, Section 4 provides new theoretical results regarding Eulerian paths, Hamiltonian paths, and vertex coloring in Delaunay Graphs. These results are complemented by the implementation of novel algorithmic methods designed to leverage these graph tools effectively. Finally, we present a case study involving the monitorization of a sugarcane field, demonstrating the practical relevance of these approaches and their performance in a real-world agricultural scenario. The structure of the paper is as follows: Section 2 introduces foundational concepts in graph theory and computational geometry.