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Let $G$ be a simple, finite, undirected graph with vertex set $\left\{v_{1}, \ldots, v_{n}\right\}$. 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 $i=1, \ldots, n$

$\operatorname{color}\left(v_{i}\right) \leftarrow \min \left\{j \in \mathbb{N}\right.$ : no neighbour of $v_{i}$ is colored $\left.j\right\}$

Which of the following statements is/are $\text{TRUE}?$

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

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Understand the Greedy Algorithm for Proper Vertex Coloring of an undirected graph: The Greedy Algorithm for Coloring Vertices

Option D is False: Greedy Algorithm Doesn't always provide Optimal Coloring

Option B is True, Option C is False: Relation between Chromatic Number & Maximum Degree

Another Example to show that the Option D is False: https://youtu.be/GKcug6W-jFA?t=4698 

Complete Analysis of this GATE 2023 question: GATE CSE 2023 - Greedy Strategy for Vertex Coloring


This COMPLETE Question, Each Option, with Proof, has been covered in GO Classes Vertex Coloring Lecture:

Graph Theory Lecture 12 - Vertex Coloring, Chromatic Number - GO Classes

GO Classes Complete Graph Theory Playlist is here:

Graph Theory Complete Course - Discrete Mathematics | GO Classes

 

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Options A and B are correct answers

Here the concept of greedy algorithm for proper vertex coloring of an undirected graph is used.
Answer:

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