## 4 Main Results
### An Algorithm for Finding the Chromatic Number of PFGs
The first type is crossing conflict, which is a collision that occurs when two separate directions of traffic try to cross paths at one spot. The second type is merging conflict, i.e., a conflict that happens when vehicles from multiple lanes or directions merge into a single lane traveling in a single direction [40]. Traffic flow is \"the number of traffic elements passing an undisturbed point upstream in the approach per unit of time\". It is measured by the number of vehicles per hour or the passenger car unit (pcu) per hour [40]. In this research, the traffic flow data are presented in pcu per hour, where the conversion factors are as follows: 0.2 for motor cycle (MC); 1 for

Figure 5: The PFG \(\hat{\mathcal{C}}\) for Example 5. Figure 6: The output of Algorithm 1 for finding \(\chi^{f}(\hat{\mathcal{C}})\) in Figure 5.

light vehicle (LV), including \"passenger cars, pick-up, and micro buses\"; and 1.3 for heavy vehicle (HV), including \"two or three-axle trucks and buses\". ### The Method to Model Traffic Flows at an Intersection Using PFGs

We visualize traffic movements from different directions as vertices and connect two vertices with an edge if the two movements are in conflict (crossing conflict or merging conflict). Hence, we obtain the data of vertex and edge sets \(V\) and \(E\), respectively. The degrees of edges and vertices show the following circumstances:

* A vertex's membership degree (\(\mu_{1}(x)\)) indicates the possibility of the crowdedness of traffic flow on the movement \(x\) at the intersection. A vertex's non-membership degree (\(\
u_{1}(x)\)) shows whether the flow of traffic on the movement \(x\) is likely to be free of congestion. The NeuM degree of a vertex (\(\eta_{1}(x)\)) indicates the possibility of an unknown circumstance about the crowdedness of traffic flow on \(x\). We obtain a PFVS \(\tilde{V}=\{(x,\mu_{1}(x),\eta_{1}(x),\
u_{1}(x))\}\). * Traffic flow on an edge that connects two vertices (traffic movements) is determined through the minimum of traffic flows on both movements. In addition, the membership degree of any edge \(uv\) in \(\tilde{E}\), that is, \(\mu_{2}(uv)\), indicates the possibility of the crowdedness of traffic flows on conflicting movements \(uv\). On the contrary, the non-membership degree \(\
u_{2}(uv)\) shows the possibility of the non-crowdedness of traffic flows on \(uv\). The NeuM degree \(\eta_{2}(uv)\) represents the possibility of the unknown condition of the crowdedness of traffic flows on \(uv\). We obtain a PFES \(\tilde{E}=\{(uv,\mu_{2}(uv),\eta_{2}(uv),\
u_{2}(uv))\}\). The degrees of vertices and edges are determined as follows:
* The membership degree \(\mu_{1}(x)\) is calculated through triangular or trapezoidal membership functions, whereas the non-membership degree is calculated through Equation (1) in Corollary 1, i.e., \(\mu_{1}(x)=\mu_{1}(f_{x})(1-\pi(x))\) and \(\
u_{1}(x)=1-\mu_{1}(x)(1-\pi(x))-\pi(x),x\in V\) by choosing \(\pi(x)=\text{mean}\{\mu_{1}(x)|x\in V\}\), where \(f_{x}\) stands for traffic flow on movement \(x\). * The NeuM degree \(\eta_{1}(x)\) is determined through the function \(g\) in Theorem 2: \[\eta_{1}(x)=g(1-\mu_{1}(x)-\
u_{1}(x))\] where \(g(x)=\frac{1-\mu_{1}(x)-\
u_{1}(x)}{a}\) for \(x\in V\) and \(a=\sum_{x\in V}(1-\mu_{1}(x)-\
u_{1}(x))\) ([35]). The NeuM degree of each edge \(\eta_{2}(uv)\) is defined similarly. * The membership degree \(\mu_{2}(uv)\) and non-membership degree \(\
u_{2}(uv)\) are calculated through formulas in Corollary 1: \(\mu_{2}(uv)=\mu_{2}(uv)(1-\pi(uv)),uv\in E\), where \(\mu_{2}(uv)=\mu_{1}(\text{min}\{f_{u},f_{v}\})\), \(\
u_{2}(uv)=1-\mu_{2}(uv)(1-\pi(uv))-\pi(uv),uv\in E\) by choosing \(\pi(uv)=\text{min}\{\mu_{2}(uv)|uv\in E\}\). ### Case Study

We take a case study at an intersection in the Special Region of Yogyakarta, Indonesia, i.e., the Pingit intersection. The location of the intersection is depicted in Figure 7 (right side).