Given a graph $G$, i have to talk about the number of ways to color this graph properly (so that no adjacent vertices have the same color). As an algorithm, i used the "Welsh-Powell" Algorithm. I have read in several references that the complexity of this algorithm for graph coloring is $O(n^2)$, but i can't find a way to prove it. Does anyone know how to show this result? Perhaps even mentioning the maximum number of colors required for this algorithm.
2026-03-31 10:58:00.1774954680
Graph coloring algorithm's complexity
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The Welsh–Powell algorithm is just the greedy algorithm where the vertices are considered in order of decreasing degree. That it is $O(n^2)$ stems from the fact that it considers each edge once when assigning a colour to a vertex. The maximum number of colours it may require is one more than the maximum degree of the graph.