Unavoidable Structures in Large and Infinite Graphs

Author

Sarah Allred, Louisiana State University and Agricultural and Mechanical CollegeFollow

Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

In this work, we present results on the unavoidable structures in large connected and large 2-connected graphs. For the relation of induced subgraphs, Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either K_r or K_r. It is well known that, for every positive integer r, every sufficiently large connected graph contains an induced subgraph isomorphic to one of K_r, K_1,r, and P_r. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. Using some techniques from the finite result, we give the unavoidable induced subgraphs of infinite 2-connected graphs. We then shift our attention to the relation of bipartite minors defined in 2016 by Chudnovsky, Kalai, Nevo, Novik, and Seymour. For the relation of bipartite minors, we present the unavoidable substructures of both large connected and large 2-connected bipartite graphs.

Date

4-3-2022

Recommended Citation

Allred, Sarah, "Unavoidable Structures in Large and Infinite Graphs" (2022). LSU Doctoral Dissertations. 5788.
https://repository.lsu.edu/gradschool_dissertations/5788

Committee Chair

Oporowski, Bogdan

DOI

10.31390/gradschool_dissertations.5788