Unavoidable Immersions and Topological Minors of k-edge-connected Graphs

Author

Brittian Qualls, 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 concerning the unavoidable structures in large and infinite k-edge-connected graphs. These results are inspired by the classical result of Ramsey, who proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either K_r or $\overline{K_r}$. We consider different graph containment relations, focusing primarily on the immersion relation. In the case of finite graphs, we provide the unavoidable immersions of 4-edge-connected graphs and prove that linear edge-connectivity suffices to immerse the graph C_t,r.

This dissertation also considers infinite graphs. We present the unavoidable topological minors of infinite 2- and 3-edge connected graphs, and we use these results to find the unavoidable immersions of graphs in these classes. We also provide results concerning the unavoidable immersions of graphs containing a vertex with infinitely many neighbors, as well as those of arbitrary (4k-1)-edge-connected graphs.

Date

3-31-2025

Recommended Citation

Qualls, Brittian, "Unavoidable Immersions and Topological Minors of k-edge-connected Graphs" (2025). LSU Doctoral Dissertations. 6720.
https://repository.lsu.edu/gradschool_dissertations/6720

Committee Chair

Ding, Guoli