Find the exact minimum vertex-color count for a small simple graph. The calculation trail makes the reported chromatic number easier to reproduce.
The chromatic number is the fewest colors needed so adjacent vertices always receive different colors.
A reproducible Graph Chromatic Number record keeps Vertices with its label and Undirected edges with its convention. The displayed Graph Chromatic Number formula then identifies the operation without guesswork.
Coloring models schedules, frequency assignment, register allocation, map constraints, and conflict separation.
Exact coloring is computationally difficult for large graphs; this page deliberately limits the vertex count. If the Graph Chromatic Number assumptions do not fit, consider maximum-color case.
Compare the result with the worked example's scale before relying on the reported chromatic number.
The sample contains a triangle, so it requires three colors even though the fourth vertex adds no new requirement. This Graph Chromatic Number example can be compared with edge representation.
Try one color, then two, and continue until backtracking finds a valid assignment.
Source labels may arrive in a different order from the form. Map them explicitly to Vertices, Undirected edges, preserve the stated ordering, and retain the defining convention for the next step.
Adding an edge can preserve or increase the chromatic number but never decrease it. Watching this response separates a data-entry mistake from an unexpected but valid value.
Check Vertices against its source before using Graph Chromatic Number. Save that Graph Chromatic Number result, change Undirected edges, and compare the revised Graph Chromatic Number case with the first.
Adjacency data defines exactly which vertex pairs are forbidden from sharing a color. This page deliberately reports only the former interpretation.
Label the output as chromatic number in notes or tables. Store vertices and undirected edges beside it when the result will be reused in a later stage.
Validate the shown coloring, then use a clique or failed lower-color search to justify minimality.
A displayed coloring proves only an upper bound until minimality is established. Cliques provide lower bounds: a triangle requires at least three colors and Kₖ requires at least k. The exact backtracking search closes the gap by confirming that every smaller color count fails for this small graph.
The chromatic number is the fewest colors needed so adjacent vertices always receive different colors.
Coloring models schedules, frequency assignment, register allocation, map constraints, and conflict separation.
Exact coloring is computationally difficult for large graphs; this page deliberately limits the vertex count.