## 8 Discussion  
In this paper, we have dealt with several algorithmic procedure regarding the existence of Eulerian or Hamiltonian trails and the vertex coloring of a Delaunay graph. Although we have not found any algorithms for this kind of graph in the literature, it is true that there are several algorithmic routines for these tools in the case of a general graph \(G\). First, [PERSON]'s and [PERSON]'s algorithms are classic approaches for finding Eulerian paths and cycles in graphs. [PERSON]'s algorithm, developed in the 19 th century, is straightforward. Figure 15: Graphs for quotients U.M./C.T. with respect to the number of sensors. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Routine** & **Complexity** & **Number of Operations** \\ \hline eulerianGraph & \(O(n^{3})\) & \(N_{1}(n)=N_{1}(n)=8+\sum\limits_{i=1}^{n}2+\sum\limits_{i=1}^{3n-6}\left(3+ \sum\limits_{i=1}^{3n-6}\left(3n(3n-6)\right)\right)\) \\ \hline createHullEdges & \(O(n)\) & \(N_{2}(n)=2+\sum\limits_{i=1}^{n-1}1\) \\ \hline findOtherEdges & \(O(n)\) & \(N_{3}(n)=1+\sum\limits_{i=1}^{3n-6}1\) \\ \hline countElements & \(O(n^{2})\) & \(N_{4}(n)=4+\sum\limits_{i=1}^{3n-6}1+\sum\limits_{i=1}^{n}\left(2+\sum\limits _{j=1}^{n}2\right)\) \\ \hline isK3 & \(O(n)\) & \(N_{5}(n)=5+2\sum\limits_{i=1}^{n}1\) \\ \hline hamiltonianGraph (1) & \(O(n^{3})\) & \(N_{6}(n)=2+\sum\limits_{i=1}^{n}(6+n(8n-18))\) \\ \hline hamiltonianGraph (2) & \(O(n^{3})\) & \(N_{7}(n)=5+\sum\limits_{i=1}^{n}\left(n^{2}+2n\right)+\sum\limits_{i=1}^{n}(5n-7)\) \\ \hline wheelCondition & \(O(n^{2})\) & \(N_{8}(n)=3+n^{2}+2\sum\limits_{i=1}^{n}2\) \\ \hline chromaticNum & \(O(n^{3})\) & \(N_{9}(n)=6+n^{3}+2\sum\limits_{i=1}^{n}2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Complexity and number of operations. ward and relies on carefully selecting edges to avoid breaking the graph into disconnected components unless no alternative exists. This method has a time complexity of \(O(e^{2})\), where \(e\) is the number of edges in the graph. This means that [PERSON]'s algorithm can reach complexity of \(O(n^{4})\) in the case of a dense graph with \(n\) vertices. In Table 5, we show that the complexity order of our routine eulerianGraph is \(O(n^{3})\) in the worst case. This means that our algorithm has a lower complexity order than [PERSON]'s. However, this is not true for [PERSON]'s algorithm, whose complexity order is \(O(e)\), i.e., \(O(n^{2})\) in the worst case. Our algorithm hamiltonianGraph in our study efficiently checks for Hamiltonian cycles and semi-Hamiltonian paths, with a complexity of \(O(n^{3})\) for dense graphs due to the need to explore all vertices, articulation points, and neighbors. This is notably more efficient than classical backtracking algorithms (whose complexity order is \(O(n! )\)) and dynamic programming approaches (where its complexity order is \(O(n^{2}\cdot 2^{n})\). Our algorithm also benefits from graph-specific optimizations (e.g., Rahman-Kaykobad's theorem and articulation point analysis), making it highly suitable for Delaunay graphs in sensor networks. This geometric focus allows it to outperform general algorithms in sparse graphs but may struggle with dense graphs, where the number of edges grows quadratically with the number of vertices, \(n\), leading to increased memory usage. Therefore, future improvements could focus on optimizing its performance in denser graphs. The algorithms in our study for computing the chromatic number in Delaunay graphs use two subroutines: wheelCondition and chromaticNum. The wheelCondition subroutine checks for isomorphism with wheel graphs to determine whether the chromatic number is 3 or 4. If the graph is not Eulerian, the chromaticNum procedure calls wheelCondition and computes the chromatic number using Maple's ChromaticNumber function, which has a complexity of \(O(n^{3})\). When compared to other algorithms, greedy algorithms (which have a complexity of \(O(n^{2})\) are faster but may not always provide the optimal solution, while backtracking methods, which have a complexity of \(O(n!