Question

Let \(G\) be a simple, finite, undirected graph with vertex set \(\{v_1, \ldots, v_n\}\). Let \(\Delta(G)\) denote the maximum degree of \(G\) and let \(\mathbb{N}=\{1,2,\ldots\}\) denote the set of all possible colors. Color the vertices of \(G\) using the following greedy strategy: for \(i = 1, \ldots, n\), \[ \text{color}(v_i) \leftarrow \min\{j \in \mathbb{N} : \ \text{no neighbour of } v_i \text{ is colored } j\}. \] Which of the following statements is/are TRUE?

• A. This procedure results in a proper vertex coloring of $G$.
• B. The number of colors used is at most $\Delta(G)+1$.
• C. The number of colors used is at most $\Delta(G)$.
• D. The number of colors used is equal to the chromatic number of $G$.

Hint:

Greedy coloring always yields a proper coloring and a universal bound of $\Delta+1$ colors, but it can be suboptimal—its performance depends on the vertex order.

Answer 1

The Correct Option is A

To solve this question, we need to analyze the properties of the given greedy coloring strategy for the graph \( G \) and verify each statement provided in the options. Let’s go through each statement one by one:

1. This procedure results in a proper vertex coloring of \( G \).
   • A proper vertex coloring ensures that no two adjacent vertices share the same color.
   • The given procedure colors each vertex \( v_i \) with the smallest possible color that none of its neighbors have. As a result, no two adjacent vertices will have the same color, since each neighbor of \( v_i \) will not be colored with the chosen color for \( v_i \).
   • Therefore, this statement is TRUE.
2. The number of colors used is at most \(\Delta(G) + 1\).
   • The greedy coloring procedure ensures that each vertex is colored with the smallest available color not used by its neighbors.
   • According to graph theory, this rule ensures that the procedure will use at most \(\Delta(G)+1\) colors for any graph. This is known as the "Greedy Coloring Theorem."
   • Therefore, this statement is TRUE.
3. The number of colors used is at most \(\Delta(G)\).
   • Typically, \(\Delta(G)\) is the maximum degree of the graph, and using only \(\Delta(G)\) colors is generally not possible for some graphs (for example, odd cycles or complete graphs). Hence, using just \(\Delta(G)\) colors doesn’t guarantee a proper coloring.
   • Therefore, this statement is FALSE.
4. The number of colors used is equal to the chromatic number of \( G \).
   • The chromatic number is the minimum number of colors needed for proper coloring.
   • The greedy algorithm can use more colors than the chromatic number, especially when the order of the vertices is not optimal for minimizing color usage.
   • Thus, this statement is FALSE.

In conclusion, the statements that are TRUE are:
1. This procedure results in a proper vertex coloring of \( G \).
2. The number of colors used is at most \(\Delta(G) + 1\).