﻿Given the wheel graph W_11 (12 nodes total: a central hub node 0 connected to all nodes 1-11, and nodes 1-11 forming a cycle 1-2-3-...-11-1):

Edges: {(0,1), (0,2), ..., (0,11), (1,2), (2,3), (3,4), (4,5), (5,6), (6,7), (7,8), (8,9), (9,10), (10,11), (11,1)}

Find a valid 4-coloring of this graph (assign each node a color from {1,2,3,4} such that no two adjacent nodes share the same color). Prove that 3 colors are insufficient by demonstrating that the odd cycle plus hub requires 4.
