Unavoidable doubly connected large graphs
A connected graph is doubly connected if its complement is also connected. The following Ramsey-type theorem is proved in this paper. There exists a function h(n), defined on the set of integers exceeding three, such that every doubly connected graph on at least h(n) vertices must contain, as an induced subgraph, a doubly connected graph, which is either one of the following graphs or the complement of one of the following graphs: (1) Pn, a path on n vertices; (2) K1,ns, the graph obtained from K 1,n by subdividing an edge once; (3) K2,n\e, the graph obtained from K2,n by deleting an edge;(4) K2,n+, the graph obtained from K2,n by adding an edge between the two degree-n vertices x1 and x2, and a pendent edge at each xi. Two applications of this result are also discussed in the paper. © 2003 Elsevier B.V. All rights reserved.
Publication Source (Journal or Book title)
Ding, G., & Chen, P. (2004). Unavoidable doubly connected large graphs. Discrete Mathematics, 280 (1-3), 1-12. https://doi.org/10.1016/j.disc.2003.05.006