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Gyárfas proved that every coloring of the edges of $$K_n$$ with $t+1$ colors contains a monochromatic connected component of size at least $n/t$. Later, Gyárfás and Sárközy asked for which values of $$\gamma=\gamma(t)$$ does the following strengthening for almost complete graphs hold: if $$G$$ is an $$n$$-vertex graph with minimum degree at least $$(1-\gamma)n$$, then every $(t+1)$-edge coloring of $$G$$ contains a monochromatic component of size at least $n/t$. We show $$\gamma= 1/(6t^3)$$ suffices, improving a result of DeBiasio, Krueger, and Sárközy.
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Unavoidable Patterns in Complete Simple Topological Graphs
We show that every complete n-vertex simple topological graph contains a topological subgraph on at least (logn)1/4−o(1) vertices that is weakly isomorphic to the complete convex geometric graph or the complete twisted graph. This is the first improvement on the bound Ω(log1/8n) obtained in 2003 by Pach, Solymosi, and Tóth. We also show that every complete n-vertex simple topological graph contains a plane path of length at least (logn)1−o(1) .
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- PAR ID:
- 10411796
- Editor(s):
- Angelini, P.
- Date Published:
- Journal Name:
- Graph Drawing and Network Visualization: 30th International Symposium, GD 2022, Tokyo, Japan, September 13–16, 2022, Revised Selected Papers
- Volume:
- 13764
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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