Graph Theory

Jim Geelen has pointed out that the proof of Lemma 21.27 is not valid for graphs with parallel edges. For example, if G is the graph obtained from the Petersen graph by doubling the edges of a perfect matching, then the 2-closure of a 2-cycle C is the subgraph H consisting of five disjoint 2-cycles. Even though this is a spanning subgraph, it is not connected. Therefore, one cannot start the induction with an arbitrary even subgraph. In the case of simple graphs, this difficulty does not arise. Nonetheless, the statement of the lemma is correct. We give here a modified proof which is valid for multigraphs.

Corrected proof of Lemma 21.27

we need to check the last sentence in the proof: “that this copy of B satisfies the required additional property”. what if the added vertex v(j) was v(2k), then the B does not necessarily have to include at least half of the vertices in the interval: v(1), v(2),….,v(2k-1).
It might be better to make the additional property of B to be related to the intervals with i = 2, 4,…,2k.

In order for figure 11.3 to correspond to the proof of the theorem, I think that every graph G23 should be changed to G13

I think that we should require that the graph has no isolated vertices, because then if it is arc-transitive it has to be both vertex and edge transitive.