# What is 3-symmetric drawing of graph

i have searched through internet, but found only paid articles. Need to understand how Petersen graph can be contracted to K33, it says what through deleting the central vertex of 3-symmetric drawig. But what is 3-symmetric drawing?

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They mean a drawing with 3-fold symmetry. The most usual drawing has 5-fold symmetry, but you should be able to track down other drawings. – Will Orrick May 7 '13 at 16:49

Here's a drawing of the Petersen graph with three-fold symmetry. With the aid of the drawing, you should be able to find the needed contraction.

It is interesting that the method suggested in your hint, which requires the deletion of a vertex and its three incident edges followed by three contractions, is not the only way to find a $K_{3,3}$ minor in the Petersen graph. An alternative requires no vertex deletions and only two edge deletions. If the graph is drawn in the usual way with five-fold symmetry, then the two deleted edges should be parallel to each other. Four contractions will then be needed.

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Your picture are very well done, could you say me which graphics editor you used? Thanks. – Yola May 8 '13 at 7:28
@Yola : I made the graph in Mathematica. I had to compute the vertex coordinates explicitly. – Will Orrick May 8 '13 at 8:58

Perhaps this should help. Here is a 3-coloring of the Petersen graph. Now contract so independent sets stay independent...

http://en.wikipedia.org/wiki/File:Petersen_graph_3-coloring.svg

In more detail, contract all blue to the neighboring green, and a pair of reds to get 3 reds and 3 greens...

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The red vertex connected to three blue vertices should be the center vertex in the diagram with three-fold symmetry. After contracting the blues to greens, don't you end up with $K_{4,3}$? I think you will then have to delete a red rather than contracting it with another red. – Will Orrick May 8 '13 at 9:02