Georgia Institute of Technology College of Sciences

Fix k vertices in a graph G, say a_1,...,a_k, if there exists a cycle that visits these vertices with this specified order, we say such a cycle is (a_1,a_2,...,a_k)-ordered. It is shown by Thomas and Wollan that any 10k-connected graph is k-linked, therefore any 10k-connected graph has an (a_1,a_2,...,a_k)-ordered for any a_1,...,a_k. However, it is possible that we can improve this bound when k is small. It is shown by W. Goddard that any 4-connected maximal planar graph has an (a_1,...,a_4)-ordered cycle for any choice of 4 vertices. We will present a complete characterization of 4-ordered cycle in planar graphs. Namely, for any four vertices a,b,c,d in planar graph G, if there is no (a,b,c,d)-ordered cycle in G, then one of the follows holds: (1) there is a cut S separating {a,c} from {b,d} with |S|\leq 3; (2) roughly speaking, a,b,d,c "stay" in a face of G with this order.