
# 15.4: Summary


• Kuratowski’s Theorem, Wagner’s Theorem
• Euler’s Formula
• $$|E| ≤ 3|V| − 6$$ for a planar graph
• Colouring maps
• The Five Colour Theorem
• The Four Colour Theorem
• Important Definitions:
• Planar graph, planar embedding
• Face
• Dual graph, planar dual
• Subdividing an edge, subdivision of a graph
• Edge contraction, contracting an edge
• Minor
• Cubic graph
• Bridge
• Snark