I’m not sure exactly what situation you’re imagining. But as an illustrative example, if the central node has eight neighbors that are connected to each other i

I’m not sure exactly what situation you’re imagining. But as an illustrative example, if the central node has eight neighbors that are connected to each other in a cycle, the graph can be coloured with three colours like this: 2 -- 3 -- 2 | \ | / | | \ | / | 3 -- 1 -- 3 | / | \ | | / | \| 2 -- 3 -- 2