¿Cómo dibujar el nombre en el centro de las caras de un gráfico?

Question

Necesito su ayuda por favor, Cómo dibujar el nombre en el centro de las caras de un gráfico, y también la cara externa, en este dibujo. Muchas gracias por vuestra ayuda comunidad. Tengo aquí el 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}    

mi dibujo se ve así ahora mismo:

El resultado deseado, por favor algo como esto:

Answer 1

mi 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 como este: