Points equidistributed over a circle, each connected to the nearest m points

         , x ~ ~ ~ - ,        
     , '               ' ,     
   ,                       ,   
  ,                         ,  
 ,                           , 
 ,                           x 
 ,                           , 
  ,                         ,  
   ,                       ,   
     ,                  , '    
       ' - x _ _ _ ,  '       

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For what values of n and m is there a path between any two points?