Greedy Coloring Algorithm

Graph Theory · Axiom Academy

LESSON Greedy Coloring Algorithm A simple and efficient heuristic for graph coloring that always produces a valid coloring The greedy coloring algorithm follows a simple strategy: The algorithm processes vertices in a predetermined order and assigns each vertex the first available color (typically starting from color 1). It never backtracks - once a vertex is colored, that color assignment is permanent. 2. Correctness: Always Produces a Valid Coloring A key property of greedy coloring is that it always produces a proper coloring - no two adjacent vertices ever receive the same color. Even though the coloring may not be optimal (using the minimum number of colors), it is guaranteed to be valid . 3. Performance Guarantee: The Delta + 1 Bound The greedy algorithm provides a theoretical guarantee on the number of colors used: chi(G) is the chromatic number (minimum colors needed) Delta(G) is the maximum degree of any vertex in G Intuition: When coloring any vertex v, at most Delta(v) colors are "forbidden" (used by neighbors). So we need at most Delta(v) + 1 colors. Since this holds for all vertices, Delta(G) + 1 colors suffice for the entire graph. 4. Vertex Ordering: Not All Orderings Are Equal The quality of the greedy coloring depends critically on the vertex ordering . Different orderings can produce dramatically different results on the same graph. 5. Example: Comparing Orderings

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