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DSATUR

DSATUR uses choosing the uncolored vertex with the highest saturation degree for vertex coloring.

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Quick Facts

Compact metadata for implementation and discovery.

Category: Graph Traversal
Difficulty: Intermediate
Family: Graph Coloring
Tier: Core
Status: Verified
Review: Source Checked
Sources: 2
Formulas: 0
Relationships: 6
Code example: Ready
Visualization: Ready
Aliases: Degree of saturation

Overview

DSATUR handles vertex coloring by choosing the uncolored vertex with the highest saturation degree.

Core Concepts

vertex coloringgraph coloringcolor refinementgraph traversal

Quick facts

Complexity

Runtime and memory behavior at a glance.

Best

O(n)

Average

O(2^n)

Worst

O(2^n)

Space

O(2^n)

When to Use

Use DSATUR when solving graph coloring and you want to benefit from choosing the uncolored vertex with the highest saturation degree.
Choose it when O(2^n) average/worst-case time and O(2^n) memory match your workload profile.
Prefer it over Greedy Graph Coloring or Welsh-Powell Coloring when the problem structure favors vertex coloring.

When Not to Use

Avoid when the graph model does not match the method's direction, weight, or connectivity assumptions.
Avoid when a simpler traversal gives the same information with less bookkeeping.

What It Does

DSATUR uses choosing the uncolored vertex with the highest saturation degree for vertex coloring
Organizes the work around vertex coloring and graph coloring.

Use Cases

Production and research workloads requiring graph coloring via choosing the uncolored vertex with the highest saturation degree.
Graph Coloring pipelines where O(2^n) time and O(2^n) space fit system constraints.
Engineering comparisons against Greedy Graph Coloring and Welsh-Powell Coloring when tuning graph coloring.

Inputs

Undirected graph G = (V, E)

Outputs

Proper vertex or edge color assignment c(x) such that adjacent elements have distinct colors

Pseudocode

1.Construct the graph representation G = (V, E) and initialize auxiliary structures for vertex coloring (graph coloring).
2.Seed the initial frontier, partition, or residual capacities according to DSATUR's invariants.
3.Iteratively process vertices or edges by choosing the uncolored vertex with the highest saturation degree.
4.Maintain structural invariants (vertex coloring, graph coloring, color refinement) and prune or contract resolved subproblems.
5.Extract and return the final graph coloring structure from the converged graph state.

Canonical Example

Input: Undirected graph G = (V, E) configured for a representative graph coloring test case
Output: Proper vertex or edge color assignment c(x) such that adjacent elements have distinct colors satisfying DSATUR invariants
Explanation: DSATUR processes the input by choosing the uncolored vertex with the highest saturation degree, tracking vertex coloring and graph coloring to produce the verified graph coloring result.

Implementation Notes

Represent vertices and edges explicitly before optimizing storage.
Keep graph direction, weights, and multiedges clear in tests.
Record tie-breaking when multiple valid outputs exist.

Common Pitfalls

Using the method on the wrong graph type, such as directed versus undirected or weighted versus unweighted.
Forgetting disconnected components or isolated vertices.
Letting tie-breaking hide nondeterministic output in tests.

Interactive Mechanism Walkthrough

Graph Frontier & Edge Relaxation

Show choosing the uncolored vertex with the highest saturation degree on a small vertex coloring example.

Step 1 of 5: Construct the graph representation G = (V, E) and initialize auxiliary structures for vertex coloring (graph coloring).

graph-traversal-mechanism

Notes

DSATUR operates in O(2^n) time and O(2^n) space within the Graph Coloring family. Often compared with Greedy Graph Coloring and Welsh-Powell Coloring.