What is a graph coloring problem?

Graph coloring assigns colors to vertices such that no two adjacent vertices share the same color, using the minimum number of colors. Chromatic number χ(G): the minimum number of colors needed. This is NP-hard in general (no polynomial algorithm for optimal coloring). Special cases: 2-colorable (Bipartite): checkable in O(V+E) with BFS/DFS — color alternately; if you need the same color for adjacent vertices, it's not bipartite; Greedy coloring: iterate vertices, assign the smallest color not used by neighbors. Not optimal but fast — uses at most Δ+1 colors (Δ = max degree); Backtracking: try each color for each vertex, backtrack if conflict — exponential worst case but works for small graphs; Welsh-Powell algorithm: sort vertices by degree descending, greedily color. Applications: (1) Register allocation in compilers (variables are nodes, conflicts are edges — minimize CPU registers needed); (2) Map coloring (four color theorem: any planar map needs ≤ 4 colors); (3) Scheduling (time slot assignment — same-time exams can't conflict); (4) Sudoku solving; (5) Frequency assignment in wireless networks. Interval graph coloring (intervals as nodes, overlapping as edges) is solvable in O(n log n) greedily.