(Since the complete graph 
  
    
      
        
          
            
              K
              
                7
              
            
          
        
      
    
    {\displaystyle {\mathsf {K_{7}}}}
   can be embedded on the torus, and 
  
    
      
        χ
        (
        
          
            
              K
              
                7
              
            
          
        
        )
        =
        7
      
    
    {\displaystyle \chi ({\mathsf {K_{7}}})=7}
  , the upper bound is tight.)
Cutting a torus
A solid torus of revolution can be cut by n (> 0) planes into maximally

  
    
      
        
          
            (
            
              
                
                  n
                  +
                  2
                
              
              
                
                  n
                  −
                  1
                
              
            
            )
          
        
        +
        
          
            (
            
              
                
                  n
                
              
              
                
                  n
                  −
                  1
                
              
            
            )
          
        
        =
        
          
            
              1
              6
            
          
        
        (
        
          n
          
            3
          
        
        +
        3
        
          n
          
            2
          
        
        +
        8
        n
        )
      
    
    {\displaystyle {\begin{pmatrix}n+2\\n-1\end{pmatrix}}+{\begin{pmatrix}n\\n-1\end{pmatrix}}={\tfrac {1}{6}}(n^{3}+3n^{2}+8n)}
  parts.The first 11 numbers of parts, for 0 ≤ n ≤ 10 (including the case of n = 0, not covered by the above formulas), are as follows:

1, 2, 6, 13, 24, 40, 62, 91, 128, 174, 230, ... (sequence A003600 in the OEIS).