# Simple vertex coloring algorithms

@article{Morris2021SimpleVC, title={Simple vertex coloring algorithms}, author={Jackson Morris and Fang Song}, journal={ArXiv}, year={2021}, volume={abs/2102.07089} }

Given a graph G with n vertices and maximum degree ∆, it is known that G admits a vertex coloring with ∆ + 1 colors such that no edge of G is monochromatic. This can be seen constructively by a simple greedy algorithm, which runs in time O(n∆). Very recently, a sequence of results (e.g., [Assadi et. al. SODA’19, Bera et. al. ICALP’20, AlonAssadi Approx/Random’20]) show randomized algorithms for ( + 1)∆-coloring in the query model making Õ(n √ n) queries, improving over the greedy strategy on… Expand

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SHOWING 1-10 OF 16 REFERENCES

Sublinear Algorithms for (Δ+ 1) Vertex Coloring

- Computer Science
- SODA
- 2019

A remarkably simple meta-algorithm for the (∆ + 1) coloring problem: Sample O(log n) colors for each vertex independently and uniformly at random from the ∆+ 1 colors; find a proper coloring of the graph using only the sampled colors of each vertex. Expand

Palette Sparsification Beyond (Δ+1) Vertex Coloring

- Computer Science, Mathematics
- APPROX-RANDOM
- 2020

This paper proves that sampling Oε(logn) colors per vertex is sufficient for proper coloring of any graph with high probability whenever each vertex is sampling from a list of (1 + ε) · deg(v) arbitrary colors, or even only deg( v) + 1 colors when the lists are the sets of sets of colors. Expand

Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models

- Mathematics, Computer Science
- ICALP
- 2020

It is proved that any randomized coloring algorithm that uses $\kappa(G)+1$ many colors, would require $\Omega(n^2)$ storage in the one pass streaming model, and many queries in the general graph query model, where $n$ is the number of vertices in the graph. Expand

Improved Quantum Algorithm for Triangle Finding via Combinatorial Arguments

- Computer Science, Mathematics
- 2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

This paper presents a quantum algorithm solving the triangle finding problem in unweighted graphs with query complexity Õ(n5/4), where n denotes the number of vertices in the graph, and shows, for the first time, that in the quantum query complexity setting un Weighted triangle finding is easier than its edge-weighted version. Expand

Quantum Query Complexity of Some Graph Problems

- Mathematics, Computer Science
- ICALP
- 2004

Almost tight lower and upper bounds for the bounded error quantum query complexity of Connectivity, StrongConnectivity, Minimum Spanning Tree, and Single Source Shortest Paths are given. Expand

Quantum algorithms for graph problems with cut queries

- Computer Science, Physics
- SODA
- 2021

It is shown that a quantum algorithm can learn a graph with maximum degree $d$ after $O(d \log(n)^2)$ many cut queries, and can learning a general graph with $O(\sqrt{m} \log (n)^{3/2})$ manycut queries. Expand

On the Power of Non-adaptive Learning Graphs

- Computer Science
- computational complexity
- 2014

A notion of the quantum query complexity of a certificate structure is introduced and there exists a function possessing the certificate structure such that a learning graph gives an optimal quantum query algorithm for it. Expand

Quantum Query Complexity for Some Graph Problems

- Computer Science
- SOFSEM
- 2004

It is proved in these cases that it is impossible to provide a better application of Ambainis’ technique for these problems, because some of the new lower bounds do not close the gap between the best upper and lower bounds. Expand

Tight bounds on quantum searching

- Computer Science, Physics
- 1996

A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table. Expand

Reducibility Among Combinatorial Problems

- Computer Science
- 50 Years of Integer Programming
- 2010

Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability. Expand