Petersen graph conjecture (Solved)

 Importance: Low ✭
 Author(s): Mkrtchyan, Vahan V. Petrosyan, Samvel S.
 Subject: Graph Theory » Basic Graph Theory » » Matchings
 Keywords:
 Posted by: vahanmkrtchyan2002 on: November 24th, 2007
 Solved by:
Conjecture   Let G be a connected bridgeless cubic graph any 2-factor of which is comprised of cycles of the length five. Then G is the Petersen graph.

not too hard

I believe that this conjecture is true, and that it can be proved with standard tools. I will sketch the argument here. I certainly wouldn't claim this is a particularly great proof.. it's just the one my nose lead me to first. The main tools I need are as follows.

Theorem  (Tutte)   If does not have a perfect matching, then there exists a subset of vertices so that has more than components which have an odd number of vertices.
Corollary   If is a connected bridgeless cubic graph, then for every edge , there exist perfect matchings so that and .

Now, let be a connected bridgeless cubic graph with the property that every 2-factor of is a (disjoint) union of cycles of length 5. Using the corollary (carefully), you can show the following in order:

\item does not have a cycle of length 2. \item does not have two triangles which share an edge. \item does not have a square and a triangle which share an edge. \item does not have a triangle. \item does not have a square.

Thus, must have girth 5. Next we will show that is 3-edge connected using the same corollary. Suppose (for a contradiction) that is only 2-edge-connected, and let be two edges which form a 2-edge cut so that and are in the same component of . Now, there must exist a perfect matching not using , so the complementary 2-factor must contain a 5-cycle which uses both the edges and . It follows that either is an edge or is an edge, and we may assume the latter without loss. Let be the neighbor of other than and let be the neighbor of other than . Now, there exists a perfect matching containing the edge , and the complementary 2-factor must contain a 5-cycle which uses all of the edges . It follows that either or , but either of these contradicts the fact that is bridgeless. This contradiction shows that is 3-edge connected. By a similar argument, you can prove that every 3-edge cut of consists of 3 edges incident with a common vertex. So, is cyclically 4-edge connected.

Next we establish that has property (*) which is a strengthening of what appears in the corollary.

(*) If and , then there is a perfect matching containing both and .

Suppose (for a contradiction) that (*) does not hold. Then has no perfect matching, so by Tutte's theorem there exists a subset of vertices so that has more than odd components. Let be the set of isolated vertices in , let be the set of odd components of with size , and let be the set of even components of . We know that by assumption, but in fact since must be an even number (as is even). Now, let and let be the edge cut in which separates from the rest of the vertices (). It follows from our construction that since every vertex in can contribute at most 3 edges to , and there are at most 6 edges in with one of as endpoint. On the other hand, since is cyclically 4-edge connected, every component in must contribute at least 4 edges to , and every vertex in contributes exactly 3 edges to , so . It follows from this that , and that every vertex in must have all three incident edges in . Thus, is a bipartite graph. Now, there exists a perfect matching of which contains the edge , and every odd cycle in the complementary 2-factor must contain either or , so the complementary 2-factor cannot have 2 odd cycles - giving us a contradiction.

With property (*) in hand, we are ready to complete the proof. Property (*) implies that every 3-edge path must be contained in a cycle of length 5, and it follows from this that every 2-edge path of is contained in at least two 5-cycles. Let be a vertex of , let be the neighbors of , and assume that the neighbors of are , , and (respectively). It follows from the fact that has girth 5 that all of these vertices we have named are distinct. Since the 2-edge path with edges is contained in two cycles of length 5, there must be edges between and . Similarly, there are edges between and , and between and . It follows from this that , and with a little more work using the girth assumption, it can be shown that is isomorphic to Petersen as required.