They're not quite the same. Certainly if there are five vertices all connected to each other then the graph can't be four-coloured. But the converse isn't true.

They're not quite the same. Certainly if there are five vertices all connected to each other then the graph can't be four-coloured. But the converse isn't true. https://standardwisdom.com/softwarejournal/2012/07/a-graph-that-requires-5-colors-but-does-not-contain-k5-complete-graph-on-5-vertices/ https://standardwisdom.com/softwarejournal/2012/07/a-graph-t...