# Lower Bounds for On-line Graph Colorings

@inproceedings{Gutowski2014LowerBF,
title={Lower Bounds for On-line Graph Colorings},
author={G. Gutowski and J. Kozik and Piotr Micek and Xuding Zhu},
booktitle={ISAAC},
year={2014}
}
We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use $$2\log _2 n - 10$$ colors, where $$n$$ is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither $$C_3$$ nor $$C_5$$ as a subgraph and forces $$\varOmega (\frac{n}{\log n}^\frac{1}{3})$$ colors. The best known on-line coloring… Expand
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#### References

SHOWING 1-10 OF 19 REFERENCES
On the minimal number of edges in color-critical graphs
The main result is that for everyk≥4 andn>k this number is at least 2, thus improving a result of Gallai from 1963 and providing some constructions of sparsek-critical graphs. Expand
On-Line and First-Fit Coloring of Graphs That Do Not Induce P5
• Mathematics, Computer Science
• SIAM J. Discret. Math.
• 1995
This article shows that there exists an on-line algorithm that will color any graph $G \in \Forb(P_{5})$ with a number of colors exponential in $\omega(G)$ and extends some of the results to larger classes of graphs defined in terms of a list of forbidden subgraphs. Expand
On-Line Coloring and Recursive Graph Theory
• Mathematics, Computer Science
• SIAM J. Discret. Math.
• 1994
The authors show that every recursive cocomparability graph can be recursively colored with a number of colors that depends only on its clique number. Expand
An on-line graph coloring algorithm with sublinear performance ratio
• Computer Science, Mathematics
• Discret. Math.
• 1989
The First-Fit algorithm has performance function n /4 and an algorithm with sublinear performance function is exhibited that is the maximum over all graphs G on n vertices of the ratio of the number of colors used by A to color G to the chromatic numbers of G . Expand
Online Coloring of Bipartite Graphs with and without Advice
• Mathematics, Computer Science
• Algorithmica
• 2013
It is proved that at least⌊1.13746⋅log2(n)−0.49887⌋ colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ⌊log2n⌓+1 for sufficiently large n. Expand
Circumference, chromatic number and online coloring
• Mathematics, Computer Science
• Comb.
• 2013
It is shown that if a graph with girth four is C5 free, then Erdős’ conjecture holds, and when the number of vertices is not too large the authors can prove better bounds on χ, and give bounds on the chromatic number of graphs. Expand
The Independence Number of Graphs with Large Odd Girth
This work improves on the lower bound of Monien and Speckenmeyer, for the independence number of a graph of order $n$ and odd girth $2k+3$ and proves similar results for graphs which are not regular. Expand
The Ramsey Number R(3, t) Has Order of Magnitude t2/log t
• Jeong Han Kim
• Mathematics, Computer Science
• Random Struct. Algorithms
• 1995
It is proved that R(3, t) is bounded below by (1 – o(1))t/2/log t times a positive constant, and it follows that R (3), the Ramsey number for positive integers s and t, has asymptotic order of magnitude t2/ log t. Expand
A Note on Ramsey Numbers
• Computer Science, Mathematics
• J. Comb. Theory, Ser. A
• 1980
Upper bounds are found for the Ramsey function and it is proved that R(3, x) cx 2 ln x and, for each k ⩾ 3, R(k,x) c k x k − 1 ( lN x) k − 2 asymptotically in x . Expand
On-line coloring k-colorable graphs. Israel
• Journal of Mathematics,
• 1998