## 4 Main Results
### Some Characteristics of the Chromatic Number of PFGs  
In this part, we investigate an upper bound for the chromatic number of PFGs in Theorem 3. **Theorem 3**: _If \(\tilde{G}=(\tilde{V},\tilde{E})\) is a PFG with the underlying graph \(G^{*}=(V^{*},E^{*})\), then_  
\[\chi^{f}(\tilde{G})\leq\max\{|N_{\tilde{E}}(x)|:x\in V^{*}\}+1.\]  
Let \(V^{*}=\{x_{1},x_{2},\cdots,x_{n}\}\). Assume \(\Gamma=\{\gamma_{1},\gamma_{2},\cdots,\gamma_{k}\}\) is a family of PF-subsets on \(\tilde{V}\) where \(\gamma_{i}=\{(x_{j},\mu_{\gamma_{i}}(x_{j}),\eta_{\gamma_{i}}(x_{j}))\},1\leq i \leq k,1\leq j\leq n\). Suppose that \(k>\max\{|N_{\tilde{E}}(x)|:x\in V^{*}\}+1\). Based on Conditions 1-2 in Definition 11, we have:  
1. \(\bigcup_{i=1}^{k}\gamma_{i}=\tilde{V}\),
2. \(\gamma_{i}\cap\gamma_{j}=\emptyset_{pfs},\forall i\
eq j\), for \(1\leq i,j\leq k\). When every pair of vertices \(x_{i},x_{j}(i\
eq j)\) is strongly adjacent, the vertices should be placed in different picture fuzzy (PF) subsets. According to the Condition 3 in Definition 11, we have \(n\) PF subsets \(\{\gamma_{1},\gamma_{2},\cdots,\gamma_{n}\}\). Meanwhile, \(\max\{N_{\tilde{E}}(x)|:x\in V^{*}\}+\ 1=n-1+1=n\). Thus, \(\chi^{f}(\tilde{G})=n=\max\{N_{\tilde{E}}(x)|:x\in V^{*}\}+1\). Otherwise, when there is a pair or weakly adjacent vertices, \(\chi^{f}(\tilde{G})=k<n=\max\{N_{\tilde{E}}(x)|:x\in V^{*}\}+1\). It is a contradiction. Thus, \(\chi^{f}(\tilde{G})=k\leq\max\{|N_{\tilde{E}}(x)|:x\in V^{*}\}+1\). Figure 2: Several cut graphs of PFG in Figure 1.  
Moreover, we investigate the connection between the chromatic number of PFGs based on strong and weak adjacencies and the \((\alpha,\beta,\delta)\)-cut chromatic number in Definition 12. Firstly, we define the chromatic number of a PFG by means of its-cut chromatic number. Definition 13: Let \(\tilde{G}=(\tilde{V},\tilde{E})\) be a PFG. The chromatic number of \(\tilde{G}\) through the \((\alpha,\beta,\delta)\)-cut chromatic number is defined as follows:  
\[\chi(\tilde{G})=\max\{\chi_{\alpha,\beta,\delta}|\alpha,\beta,\delta\in L\},\]  
where \(L\) is the level set of \(\tilde{G}\) and \(\chi_{\alpha,\beta,\delta}=\chi(G_{\alpha,\beta,\delta})\). When \(\delta=0\), Definition 13 becomes the chromatic number of IFGs. In certain conditions, we prove that the chromatic number of the underlying graph of PFG is equal to the chromatic number concept in Definition 13. Example 3: Let us consider Example 2. According to Definition 13, the chromatic number of \(\tilde{G}\) through the \((\alpha,\beta,\delta)\)-cut is  
\[\chi(\tilde{G})=\max\{\chi_{\alpha,\beta,\delta}|\alpha,\beta,\delta\in L\}=3.\]  
Theorem 4: _Given a PFG \(\tilde{G}=(\tilde{V},\tilde{E})\) with the underlying graph \(G^{*}=(V^{*},E^{*})\). If \(\alpha_{1}=\min\{\alpha|\alpha\in L\}\), \(\beta_{1}=\min\{\beta|\beta\in L\}\), and \(\delta_{1}=\max\{\delta|\delta\in L\}\), then the chromatic number of \(\tilde{G}\):_  
\[\chi(\tilde{G})=\chi(G^{*}).\]  
Proof: Since \(\alpha_{1}=\min\{\alpha|\alpha\in L\}\), \(\beta_{1}=\min\{\beta|\beta\in L\}\), and \(\delta_{1}=\max\{\delta|\delta\in L\}\), we have \(\chi(G_{\alpha_{1},\beta_{1},\delta_{1}})=\ \ \max\{\chi_{\alpha,\beta, \delta}|\alpha,\beta,\delta\in L\}\). Further, all vertices and edges of \(\tilde{G}\) become elements of the crisp graph \(G_{\alpha_{1},\beta_{1},\delta_{1}}\). This implies \(G_{\alpha_{1},\beta_{1},\delta_{1}}=G^{*}\) and \(\chi(G^{*})=\chi(G_{\alpha_{1},\beta_{1},\delta_{1}})=\max\{\chi_{\alpha, \beta,\delta}|\alpha,\beta,\delta\in L\}=\chi(\tilde{G})\). Theorem 5: _Let \(\tilde{G}=(\tilde{V},\tilde{E})\) be a PFG with the underlying graph \(G^{*}=(V^{*},E^{*})\). If all edges in \(\tilde{E}\) connect strongly adjacent vertices, then_  
\[\chi^{f}(\tilde{G})=\chi(G^{*}).\]  
Proof: Assume that \(\chi^{f}(\tilde{G})=k\). According to Definition 11, we have a family \(\Gamma=\{\gamma_{1},\gamma_{2},\dots,\gamma_{k}\}\) of PF-subsets of \(\tilde{V}\) such that it satisfies 3 conditions in Definition 11:  
1. \(\bigcup_{i=1}^{k}\gamma_{i}=\tilde{V}\),
2.