On probabilistic, planar, and descriptive graph coloring problems
Author(s)
Anderson, James
Advisor(s)
Bernshteyn, Anton
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Abstract
We investigate graph coloring problems from the perspective of three different types of graphs: graphs with forbidden bipartite subgraphs; planar graphs; and Borel graphs that are line graphs.
We start with studying graph with forbidden bipartite subgraphs. We say a graph is $F$-free if it contains no subgraph (not necessarily induced) isomorphic to graph $F$. It follows from recent work by Davies, Kang, Pirot, and Sereni that if $G$ is $K_{t,t}$-free, then $\chi(G) \leq (t + o(1)) \Delta / \log\Delta$ as $\Delta \to \infty$. We improve this bound to $(1+o(1)) \Delta/\log \Delta$, making the constant factor independent of $t$. We further extend our result to the correspondence coloring setting.
Next we study defective coloring of planar graphs. For $d \in \mathbb{N}$, a coloring of a graph is \textit{$d$-defective} if every vertex is colored the same as at most $d$ of its neighbors. We investigate defective coloring of planar graphs in the context of correspondence coloring. First we show there exist planar graphs that are not $3$-defective $3$-correspondable, strengthening a recent result of Cho, Choi, Kim, Park, Shan, and Zhu. Then we construct a planar graph that is $1$-defective $3$-correspondable but not $4$-correspondable, thereby extending a recent result of Ma, Xu, and Zhu from list coloring to correspondence coloring. Finally we show all outerplanar graphs are $3$-defective 2-correspondence colorable, with $3$ defects being best possible.
Finally, we study Borel graphs. We characterize Borel line graphs in terms of 10 forbidden induced subgraphs, namely the 9 finite graphs from the classical result of Beineke together with a 10th infinite graph associated to the equivalence relation $\mathbb{E}_0$ on the Cantor space. As a corollary, we prove a partial converse to the Feldman--Moore theorem, which allows us to characterize all locally countable Borel line graphs in terms of their Borel chromatic numbers.
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Date
2024-12-02
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Dissertation