Degree

Doctor of Philosophy (PhD)

Department

Mathematics

Document Type

Dissertation

Abstract

The goal of this thesis is to determine the unavoidable topological minors of large and infinite $ 3 $-connected rooted graphs, where a rooted graph is a graph $ G $ together with a specified subset $ X $ of $ V(G) $ or $ E(G) $. We have two results for finite graphs. First, every $ 3 $-connected finite graph $ G $ with a sufficiently large $ X \subseteq E(G) $ must contain a topological minor $ K_{3, n}, W_n $, or $ V_n $, using many edges of $ X $, where $ W_n $ is a wheel with $ n $ spokes and $ V_n $ is obtained from a ladder with $ n $ rungs by adding two grips and a handle. Second, every $ 3 $-connected finite graph $ G $ with a sufficiently large $ X \subseteq V(G) $ must contain a topological minor $ K_{3, n}, K_{3, n}^1, K_{3, n}^2, K_{3, n}^3, W_n $, or $ V_n $, using many vertices of $ X $, where $ K_{3, n}^i \; (i = 1, 2, 3) $ is obtained by gluing the leaves of $ i $ combs and $ 3 - i $ stars in the natural way.

We also have two results for infinite graphs. First, every $ 3 $-connected graph $ G $ with an infinite $ X \subseteq E(G) $ must contain a topological minor $ K_{3, \infty}, FF, FL $, or $ LL $, using infinitely many edges of $ X $, where $ FF, FL $, and $ LL $ are obtained, respectively, by gluing $ i \; (i = 0, 1, 2) $ infinite ladders and $ 2 - i $ infinite fans along their rails. Second, every $ 3 $-connected graph $ G $ with an infinite $ X \subseteq V(G) $ must contain as a subgraph a subdivision of $ K_{3, \infty}, FF, FL $, or $ LL $, containing infinitely many vertices of $ X $. We also discuss similar results for lower connectivities, which in fact are corollaries of results listed above.

Date

4-4-2024

Committee Chair

Ding, Guoli

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