## 4 Main Results
### An Algorithm for Finding the Chromatic Number of PFGs
Tentara Pelajar street is in the Southern (S) direction, Diponegoro street is to the Eastern (E) direction, Magelang street is to the Northern (N) direction, and Kyai Mojo street is to the Western (W) direction. A sketch of the intersection is also given in Figure 7 (left side). We collect the data of traffic flows on 25-27 January 2023 in the morning (06.30-07.30 a.m.) and in the evening (16.30-17.30 p.m.). There are 12 traffic movements in the intersection, i.e., WN, WE, WS, SN, SW, SE, NE, NW, NS, EN, EW, and ES. This means that the vertex set \(V\) contains 12 vertices. The next step is to transform the traffic flow data into PFVS \(\tilde{V}\), wherein the membership degree of each element is determined via triangular and trapezoidal membership functions in Figure 8.

It is shown in Table 1 that traffic flows on WN and EW have a high possibility of being crowded compared to traffic flows on other movements. Conversely, the traffic flow on SW has a lower possibility of being crowded. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Traffic Movements** & **Traffic Flows** & **Clusters** & **Degree of Vertices** \\ \hline WN & 1805 & High & (0.383, 0.083, 0) \\ WE & 605 & Medium & (0.262, 0.083, 0.121) \\ WS & 100 & Low & (0.306, 0.083, 0.077) \\ SN & 354 & Low & (0.112, 0.083, 0.271) \\ SW & 32 & Low & (0.359, 0.083, 0.024) \\ SE & 458 & Low & (0.032, 0.083, 0.351) \\ NE & 530 & Medium & (0.166, 0.083, 0.217) \\ NW & 831 & Medium & (0.216, 0.083, 0.167) \\ NS & 838 & Medium & (0.207, 0.083, 0.176) \\ EN & 337 & Low & (0.125, 0.083, 0.258) \\ EW & 1210 & High & (0.339, 0.083, 0.044) \\ ES & 742 & Medium & (0.329, 0.083, 0.054) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The PFVS \(\tilde{V}\) of traffic flow data in the Pingit intersection. Figure 8: The function to calculate membership degree in the PFVS \(\tilde{V}\). Figure 7: Location of the Pingit intersection in the Special Region of Yogyakarta. ### Comparison to the Fuzzy Graph Coloring Method

In this part, we compare the result in Table 4 with a traffic signal phasing obtained from the fuzzy graph coloring method based on \(\delta\)-fuzzy independent vertices (\(\delta\in[0,1]\)) as given in [33]. The fuzzy graph model of traffic flows in the Pingit intersection is depicted in Figure 11.

For \(\delta=0.8857\), the sets \(\{SW,EW,NW\}\), \(\{ES,WS,NS\}\), and \(\{SE,WE,NE\}\) are the sets of \(\delta\)-fuzzy independent vertices since \(\mu(uv)\leq\delta\) for each pair \(u,v\) in the above sets. Therefore, we obtain 4-phase scheduling as follows:

\[(i)SW,EW,NW,WN;\quad(ii)SN,NS,ES;\quad(iii)EN,WS,NE;\quad(iv)WE,SE. \tag{2}\]

We observe that some pairs of vertices in (2) are elements of merging conflict in Table 3. Hence, the traffic signal phasing obtained from PFG coloring in Table 4 is safer than the signal phasing from the fuzzy graph coloring in (2) since there are no traffic flows from merging conflict that move simultaneously at the same phase. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Patterns** & **Phase 1** & **Phase 2** & **Phase 3** & **Phase 4** \\ \hline
1 & WN, WE, WS & SE, SN, SW & NE, NW, NS & EN, EW, ES \\
2 & WE, WS, SW & SN, SE, ES & WN, NE, NW, NS & EN, EW \\
3 & WS, SW, EN & SE, NW, ES & SN, NE, NS & EW, WE, WN \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The traffic signal phasing that can be implemented at the Pingit intersection. Figure 10: Determination of the chromatic number of PFG model \(\bar{\mathcal{G}}\) in Figure 9.

## 6 Conclusions

The concept of strong and weak adjacencies between vertices could be implemented in making decisions regarding real-world problems. Therefore, we generalized the concept from the intuitionistic fuzzy graph (IFG) into the picture fuzzy graph (PFG) in the previous work. In this research, we investigated some of the characteristics of the chromatic number of PFGs based on strong and weak adjacencies between vertices and their relation to the \((\alpha,\beta,\delta)\)-cut chromatic numbers.