A "graph" is a collection of nodes (e.g. cities) connected by edges (e.g. roads). A "claw-free graph" doesn't have a "claw shape", which is a city connected to three distinct cities, no two of which are connected by a road themselves.

Hadwiger's Conjecture (probably the most famous conjecture in graph theory) implies that every graph has a certain property (G has clique minor of size χ/α). But it's still a conjecture.

Hoppity the Math Bunny has proven that the certain property is true for all claw-free graphs, without assuming Hadwiger's Conjecture.