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I'm pretty sure that triangles and 3-faces are not the same but I cannot find their differences according to their definitions. Could you please help me with that?

Thanks

arny
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1 Answers1

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In a planar graph the two notions are equivalent, but the notion of a triangle is more general: a $3$-clique (i.e., a copy of $K_3$) in any graph is a triangle, but the notion of face applies only to planar graphs.

Brian M. Scott
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  • Thanks. But I heard something like this: If a triangle has some other edges inside it's not 3-face. In other words, any 3-faces are triangles but all triangles are not 3-faces. Does it make sense? – arny Jun 23 '15 at 20:03
  • @arny: In a planar graph a triangle can’t have any edges inside. If the vertices of the triangle are $u,v$, and $w$, any edge inside the triangle would have to have two of $u,v$ and $w$ as endpoints and would therefore just be one edge of the triangle. – Brian M. Scott Jun 23 '15 at 20:05