## 4 Main Results
### An Algorithm for Finding the Chromatic Number of PFGs  
We create an algorithm to compute the chromatic number of PFGs with the exception for CPFG. We use the assumption that not all edges in \(\tilde{E}\) connect strongly adjacent vertices. Algorithm 1 can also be used for the coloring of IFGs when the inputs are the intuitionistic fuzzy vertex set and the intuitionistic fuzzy edge set. ```
0: The PFG \(\tilde{\mathcal{G}}=(\tilde{\mathcal{V}},\tilde{E})\) with the elements: * Vertex set \(V=\{v_{1}\},i=1,2,\cdots,n\), and edge set \(E=\{e_{1}\},l=1,2,\cdots,m\). * Degree of vertices \(W_{V}=\{w_{v_{i}}\}\), with \(w_{v_{i}}=(\mu_{1}(v_{i}),\eta_{1}(v_{i}),\
u_{1}(v_{i}))\). * Degree of edges \(W_{E}=\{w_{v_{i}}\}\), where \(w_{v_{i}}=(\mu_{2}(e_{1}),\eta_{2}(e_{i}),\
u_{2}(e_{i}))\), \(e_{i}=v_{i}v_{i}\
eq j\in\{1,\cdots,n\}\). 0: The chromatic number: \(\chi^{\prime}(\tilde{\mathcal{G}})\)
1:for\(h=1\) to \(n-1\)do
2:for\(j=1\) to \(n-h\)do
3: Check for all pairs \(Q_{h}=[v_{h}v_{j+k}]\)
4:if\(\frac{1}{2}\min(\mu_{1}(v_{h}),\mu_{1}(v_{j+h}))\leq\mu_{2}(v_{h}v_{j+h})\), \(\frac{1}{2}\min(\eta_{1}(v_{h}),\eta_{1}(v_{j+h}))\leq\eta_{2}(v_{h}v_{j+h})\), and \(\frac{1}{2}\max(v_{1}(v_{h}),v_{1}(v_{j+h}))\leq v_{2}(v_{h}v_{j+h})\)then
5:\(\ \We show that Algorithm 1 gives the chromatic number \(\chi^{f}(\tilde{G})=k\) for \(k\leq n\) and proves the correctness of the algorithm by using mathematical induction on the cardinality \(|V|\) as follows. **Base step**: for \(|V|=2\) and \(V=\{v_{1},v_{2}\}\). If the two vertices in \(V\) are strongly adjacent vertices, then Steps 1-11 will produce \(C_{1}=\emptyset\). Further, Steps 12-16 give \(\gamma_{11}=\{(v_{1},\mu_{1}(v_{1}),\eta_{1}(v_{1}),\
u_{1}(v_{1}))\}\), \(\gamma_{12}=\{(v_{2},\mu_{1}(v_{2}),\eta_{1}(v_{2}),\)\(v_{1}(v_{2}))\}\), and the chromatic number \(\chi^{f}(\tilde{G})=2\). If the two vertices in \(V\) are weakly adjacent vertices, then Steps 1-11 produce \(C_{1}=\{v_{1},v_{2}\}\). Further, Steps 17-36 give PF-subset \(\gamma_{1}=\{(v_{1},\mu_{1}(v_{1}),\eta_{1}(v_{1}),\
u_{1}(v_{1})),(v_{2},\mu_{ 1}(v_{2}),\eta_{1}(v_{2}),\
u_{1}(v_{2}))\}\). Finally, Steps 47-48 give the family \(\Gamma=\{\gamma_{1}\}\) and \(\chi^{f}(\tilde{G})=1\). Thus, the chromatic number \(\chi^{f}(\tilde{G})=k\leq 2\). The base step is satisfied. **Inductive step**: Assume that Algorithm 1 is correct for cardinality \(|V|=n\). Steps 1-11 produce the set of weakly adjacent vertices and the set of strongly adjacent vertices. Furthermore, Steps 20-41 give the PF-subset:

\[\gamma_{11},\gamma_{12},\cdots,\gamma_{1,k1};\gamma_{21},\gamma_{22},\cdots, \gamma_{2,k2};\cdots;\gamma_{s1},\gamma_{s2},\cdots,\gamma_{s,ks}.\]

In Steps 42-48, we obtain \(\Gamma=\{\gamma_{1},\gamma_{2},\cdots,\gamma_{k}\}\), where \(k=k_{min}=\min\{k1,k2,\cdots,ks\}\) and \(\gamma_{1}=\gamma_{k_{min}1},\gamma_{2}=\gamma_{k_{min}2},\cdots,\gamma_{k}= \gamma

**Example 5**. : _Let us consider PFG \(\tilde{G}=(\tilde{V},\tilde{E})\) in Figure 5, where \(\tilde{V}\) is a PFS on \(V=\ \{WE,WS,\ SN,SE,NW,NS,EN,EW\}\)._

_The output of Algorithm 1 for PFG \(\tilde{G}\) in Figure 5 is shown in Figure 6. We obtain the family \(\Gamma=\ \{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4}\}\) and the chromatic number \(\chi^{f}(\tilde{G})=\mathbf{4}\)._

Figure 4: The output of Algorithm 1 for finding \(\chi^{f}(\tilde{G})\) in Figure 3.

Figure 3: The picture fuzzy graph \(\tilde{G}\) for Example 4. ## 5 Experimental Result

In this section, we discuss an implementation of Algorithm 1 in determining traffic signal phasing at an intersection. A phase is defined as any traffic light display that has its own timings, and it determines how a specific vehicle or pedestrian will move, whereas the term phase set refers to \"any distinct combination of concurrent vehicle or pedestrian phases\". Conflicting phases are those that cannot both have green indicators at the same time [39]. There are two types of conflicts between traffic movements at an intersection.