Finding Chromatic Numbers

Graph Theory · Axiom Academy

EXAMPLE Finding Chromatic Numbers Determine the chromatic number of a graph using cliques and colorings Graph G with 6 vertices (wheel graph W 5 ) Excellent work! You've successfully determined the chromatic number. Here's what we learned: Chromatic Number Definition: χ(G) is the minimum number of colors needed to properly color all vertices such that no two adjacent vertices share the same color. Lower Bound Strategy: Finding a clique (complete subgraph) gives us a lower bound: χ(G) ≥ ω(G), where ω(G) is the clique number. Every vertex in a clique needs a different color. Upper Bound Strategy: Constructing an actual valid coloring gives us an upper bound. If we can color the graph with k colors, then χ(G) ≤ k. Proving Tightness: When the lower and upper bounds match, we've found the exact chromatic number. In this case, χ(G) = 3 because we proved 3 ≤ χ(G) ≤ 3. This systematic approach works for any graph: find cliques for lower bounds, construct colorings for upper bounds, and refine until the bounds converge!

This is the written version of the interactive lesson above. See the full Graph Theory course.