Como desenhar o nome no centro das faces de um gráfico?

Question

Preciso da sua ajuda por favor, Como desenhar o nome no centro das faces de um gráfico, e a face externa também, neste desenho. Muito obrigado pela sua comunidade de ajuda. Tenho aqui o código:

    \begin{tikzpicture}
        % Source : https://www.baeldung.com/cs/latex-drawing-graphs 
        % https://tex.stackexchange.com/questions/270543/draw-a-graph-in-latex-with-tikz
        % https://tex.stackexchange.com/questions/315456/colouring-graphs-tikz      
        % source ellipse : https://www.overleaf.com/learn/latex/LaTeX_Graphics_using_TikZ%3A_A_Tutorial_for_Beginners_(Part_1)%E2%80%94Basic_Drawing
        \draw  [fill = green!5!white]  (2,2) ellipse (7cm and 4cm);
        
        %\draw  [fill = green!5!white] 
        %       (-1,-2) .. controls (0,-3) .. (5,-2) .. controls (9,1) and (7,7) .. (4,7.5) .. controls (0,8) and (-4,8) .. (-1,-2) ;                                   
        
        \path
        ( 0  , 3 ) coordinate (1)
        ( 2.5, 4 ) coordinate (2)
        ( 5  , 3 ) coordinate (3)
        ( 3.5, 1 ) coordinate (4)
        ( 1  , 0 ) coordinate (5);      
        
        \draw [fill = violet] (1) to [out=125,in=90,looseness=2] (3);
        \draw [fill = yellow] (1) to [out=-120,in=-90,looseness=3] (4);     
        \draw (1) -- (3) ;  
        \draw [fill = orange] (2) -- (4) -- (5) -- (2);
        \draw [fill = cyan] (2) -- (3) -- (4) -- (2); 
        \draw [fill = pink] (1) -- (2) -- (5) -- (1);
        \draw (5) -- (1);
        \draw (2) -- (4);   
        \draw (2) -- (5);       
        
        \tikzstyle{vertex} = [draw,circle,fill=white,inner sep = 3pt]
        \node[vertex] [label = left: 1] at (1) {};
        \node[vertex] [label = above : 2] at (2) {};
        \node[vertex] [label = below: 3] at (3) {};
        \node[vertex] [label = right: 4] at (4) {};
        \node[vertex] [label = left: 5] at (5) {};
        %   \draw (0,0) arc (0:270:8mm);
        %   \draw (0,0) arc (0:315:1.75cm and 1cm); 
    \end{tikzpicture}    

meu desenho está assim agora:

O resultado desejado, por favor, algo assim:

Answer 1

meu resultado final

\begin{tikzpicture}
    % Source : https://www.baeldung.com/cs/latex-drawing-graphs 
    % https://tex.stackexchange.com/questions/270543/draw-a-graph-in-latex-with-tikz
    % https://tex.stackexchange.com/questions/315456/colouring-graphs-tikz      
    % source ellipse : https://www.overleaf.com/learn/latex/LaTeX_Graphics_using_TikZ%3A_A_Tutorial_for_Beginners_(Part_1)%E2%80%94Basic_Drawing
    \draw  [fill = green!5!white]  (2,2) ellipse (7cm and 4cm);
    
    %\draw  [fill = green!5!white] 
    %       (-1,-2) .. controls (0,-3) .. (5,-2) .. controls (9,1) and (7,7) .. (4,7.5) .. controls (0,8) and (-4,8) .. (-1,-2) ;                                   
    
    \path
    ( 0  , 3 ) coordinate (1) 
    ( 2.5, 4 ) coordinate (2)
    ( 5  , 3 ) coordinate (3)
    ( 3.5, 1 ) coordinate (4)
    ( 1  , 0 ) coordinate (5);      
    
    \draw [fill = violet] (1) to [out=125,in=90,looseness=2] (3);
    \draw [fill = yellow] (1) to [out=-120,in=-90,looseness=3] (4);     
    \draw (1) -- (3) ;  
    \draw [fill = orange] (2) -- (4) -- (5) -- (2);
    \draw [fill = cyan] (2) -- (3) -- (4) -- (2); 
    \draw [fill = pink] (1) -- (2) -- (5) -- (1);
    \draw (5) -- (1);
    \draw (2) -- (4);   
    \draw (2) -- (5);       
    
    \tikzstyle{vertex} = [draw,circle,fill=white,inner sep = 3pt]
    \node[vertex] [label = left: 1] at (1) {};
    \node[vertex] [label = above : 2] at (2) {};
    \node[vertex] [label = below: 3] at (3) {};
    \node[vertex] [label = right: 4] at (4) {};
    \node[vertex] [label = left: 5] at (5) {};
    %   \draw (0,0) arc (0:270:8mm);
    %   \draw (0,0) arc (0:315:1.75cm and 1cm); 
    
    \path
    (-2 , 3  ) coordinate () {} node[left]      {face6}
    (1.5, 4.5) coordinate () {} node[left]      {face1}
    (1.8, 2.5  ) coordinate () {} node[left]    {face2}
    (3, 1.5) coordinate () {} node[left]        {face3}
    (4.5, 2.5) coordinate () {} node[left]      {face4}
    (2.5, -0.7) coordinate () {} node[left]     {face5}
    ;
    
\end{tikzpicture}   
\\ \\ \\
 \begin{tikzpicture}[c/.style={circle,draw,inner sep=2pt}] 
    \path[nodes={text=teal}]  
    (0,0)   node[c] (v1) {} node[below left]    {$v_1$} 
    (15,0)  node[c] (v2) {} node[below right]   {$v_2$} 
    (10,1)  node[c] (v3) {} node[below]         {$v_3$} 
 %  (8,2)   node[c] (v3) {} node[below]         {$v_4$} 
    (7,8)   node[c] (v16){} node[above]         {$v_{16}$} ; 
    
    
    \draw (v1)--(v2)--(v3)--(v1); 
    \draw (v1) to[bend left=30] (v16) (v2) to[bend right] (v16) ; 
\end{tikzpicture}

Resultado assim: