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 10×12 = 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 Collections: Mathematics