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Mathematics

Chromatic Number and Clique Number

Quick fact

There are graphs that contain no triangle (no clique of size 3) yet require over 1,000 colors to properly color! Such triangle-free graphs with arbitrarily large chromatic number were first constructed by Mycielski in 1955, showing the clique number alone cannot bound the coloring requirement.

Why this is interesting

You've probably seen puzzles about coloring maps so neighboring countries get different colors. But what if we color graphs instead? The minimum number of colors needed is called the chromatic number—and the biggest cluster of all-connected vertices, the clique number, seems to set a floor. Yet sometimes the floor is far from the truth.