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Math 4140 Homework 3 (Due: Friday Feb. 17th) Problem 1 Find the automorphism group of the Petersen graph. You must prove your answer
 

Summary: Math 4140 Homework 3 (Due: Friday Feb. 17th)
Problem 1 Find the automorphism group of the Petersen graph. You must prove your answer
is correct. Hint: first used the orbit stabilizer lemma to get the size then use a presentation of
the graph given in the first week of class to help you nail down the group.
Solution: The drawing on the left above is the presentation of the Petersen graph with vertices
that are 2-subsets of {0, 1, 2, 3, 4} and edges between vertices that are disjoint as sets. If we
apply any permutation S5 to the set {0, 1, 2, 3, 4} then it renames the vertices in a manner
that preserves disjointness. This means that each such permutation is an automorphism of the
Petersen graph. Different permutations take at least one point to a different destination and so
take pairs of points to different destinations giving us 120 automorphisms of the Petersen graph.
Notice also that the action of these 120 automorphisms can take any vertex to any other, telling
us that the orbit of any vertex has size ten. Consider the stabilizer S in the automorphism group
of the center point in the drawing above right; we know from class that this is a drawing of the
Petersen graph. Then the three vertices adjacent to the central vertex are on `diagonals' of the
displayed hexagon. Since a symmetry of the hexagon takes diagonals to diagonals id follows all
the 12 symmetries of the hexagon in D6 are paired with symmetries in S telling us that |S| = 12.
The orbit stabilizer lemma then tells us the size of the automorphism group is 1012 = 120 and
so the automorphisms induced by S5 are all of them and the automorphism group is isomorphic
to S5.
Problem 2 Find the chromatic number of the generalized Petersen graph Pn,m. The problem is

  

Source: Ashlock, Dan - Department of Mathematics and Statistics, University of Guelph

 

Collections: Mathematics