## 6 Case Study: Monitorization of a Sugar Cane Field  
All the algorithms proposed in the previous section can be used to manage the devices in an agricultural field. This is the case of the situation presented here, in which we show two examples for the monitorization of two soil variables--moisture and temperature--of a sugarcane field that were needed to control the irrigation. The monitorization network is built with a set of devices, called nodes, which have different capabilities: measurement of soil moisture and temperature; computation, to execute some local filters and routines, and storing; and communication of the processed data using the LoRaWAN protocol. The heart of each node is a Lopy4 board from Pycom, whose core is an ESP32 processor. A photograph of the node is shown in Figure 8.  
In the first example, a set of 12 nodes was deployed in a 100 \(\times\) 30 meters field close to the city of Arroyos y Esteros in Paraguay (Longitude 57\({}^{\circ}\)04\({}^{\prime}\)50\({}^{\prime\prime}\) West; Latitude 25\({}^{\circ}\)05\({}^{\prime}\)40\({}^{\prime\prime}\) South). The location of the nodes is depicted in Figure 9, where the \(x\)-axis is aligned with the direction of the sugarcane planks. Now, we consider the 12 nodes from Figure 9 and define the variable PTS containing the Euclidean coordinates of each node. PTS:=[[5,13],[10,6],[25,28],[30,20],[35,3],[45,11],[55,16],[60,29],[75,22],
[80,5],[85,28],[95,12]];  
In Figure 10, we show the Voronoi diagram and weighted Delaunay graph associated with the nodes. Now, we are going to apply the different algorithms in the Delaunay graph created from the sugarcane field. First, when we run the routine eulerianGraph, we obtain the output  
\"The graph isn't Eulerian or semi-Eulerian\"  
This is due to the fact that our Delaunay graph contains more than two vertices with odd degrees. If we apply the algorithmic procedures based on the Lemma 1, createHullEdges, findOtherEdges, countElements, and isK3, we obtain:  
\"Is Eulerian? :\"  
\"No, all the intern edges do not create K3\"  
Figure 8: Deployed node in a sugarcane field. As a consequence of this, it is not possible to design a trail in the sensor network that visits every communication link once. Next, we execute the routine called hamiltonianGraph obtaining:  
The graph is Hamiltonian  
The Hamiltonian cycle is \([1,2,4,5,6,7,9,10,12,11,8,3,1]\)  
Consequently, it is possible to visit all sensors once following a closed trail in order to collect all the data saved by them. The sequence obtained gives us the node order that should be followed. Finally, we analyze some properties of our sensor network related to the chromatic number using wheelCondition and chromaticNum. The outputs obtained are as follows:  
\"The graph is not Eulerian\"  
\"The graph or a subgraph is a Wheel Graph\", \(W_{6}\)  
\"The wheel graph is even\"  
\"The chromatic number must be 4\"  
\"The chromatic number is:\", 4  
\[[3,5,7,12],[2,6,9],[1,8,10],[4,11]]\]  
Figure 10: Voronoi diagram and weighted Delaunay graph associated with the sugarcane field. Figure 9: Aerial photography of the sugarcane field with the nodes marked with blue dots. In this way, the chromatic number of our Delaunay graph associated with the sugar cane field is 4 and we also know the color of each vertex. This allows us to assign different tasks or roles to adjacent nodes in a minimal way. The first task would be assigned to nodes \(3,5,7\), and \(12\). Then, the second one would be assigned to vertices \(2,6\) and \(9\), the third one to \(1,8\) and \(10\), and the last one to nodes \(4\) and \(11\). Another possible application is to avoid interference between adjacent vertices by developing the previous assignment considering channels instead of tasks. In case of energy consumption issues, we can activate the group of sensors associated with one color and deactivate the other ones. It could be an efficient way of saving energy since the communication links are not working since there is no pair of adjacent nodes activated.