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Let G be a simple, finite, undirected graph with vertex set {v1,..., vn}. Let Δ(G) denote the maximum degree of G and let N = {1, 2,...} denote the set of all possible colors. Color the vertices of G using the following greedy strategy: for i = 1,..., n, color(vi) ← min{j ∈ N: no neighbour of vi is colored j}. Which of the following statements is/are TRUE?

1. This procedure results in a proper vertex coloring of G.
2. The number of colors used is at most Δ(G) + 1.
3. The number of colors used is at most Δ(G).
4. The number of colors used is equal to the chromatic number of G.

Correct answer: The number of colors used is at most Δ(G) + 1.

Solution

The greedy coloring strategy ensures that each vertex is assigned the smallest color not used by its neighbors, which can require at most one additional color beyond the maximum degree of the graph, thus guaranteeing that the number of colors used does not exceed Δ(G) + 1.

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