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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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Notes on Growing a Tree in a Graph
We study the height of a spanning tree $$T$$ of a graph $$G$$ obtained by starting with a single vertex of $$G$$ and repeatedly selecting, uniformly at random, an edge of $$G$$ with exactly one endpoint in $$T$$ and adding this edge to $$T$$.
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- Award ID(s):
- 1661063
- PAR ID:
- 10158516
- Date Published:
- Journal Name:
- Random structures algorithms
- Volume:
- 55
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- 290-312
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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