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?

Let us analyze the given greedy coloring procedure step-by-step:

1. Statement (A) is TRUE: At each step $i$, the vertex $v_i$ is assigned the smallest color $j \in \mathbb{N}$ that has not been assigned to any of its adjacent vertices that have already been colored. Consequently, no two adjacent vertices ever receive the same color, guaranteeing a proper vertex coloring of $G$.

2. Statement (B) is TRUE: The maximum degree of any vertex in $G$ is $\Delta(G)$. Therefore, each vertex $v_i$ has at most $\Delta(G)$ neighbors. When vertex $v_i$ is being colored, at most $\Delta(G)$ distinct colors are unavailable. In the set of the first $\Delta(G) + 1$ colors, $\{1, 2, \ldots, \Delta(G) + 1\}$, there must always be at least one available color. Hence, the greedy coloring algorithm will use at most $\Delta(G) + 1$ colors.

3. Statement (C) is FALSE: For a complete graph $K_n$, the maximum degree is $\Delta(K_n) = n - 1$. Since every pair of vertices is connected, $n = \Delta(K_n) + 1$ colors are required. Thus, the greedy algorithm will use $\Delta(G) + 1$ colors, which is strictly greater than $\Delta(G)$.

4. Statement (D) is FALSE: The number of colors used by greedy coloring depends heavily on the ordering of the vertices and is not guaranteed to achieve the chromatic number $\chi(G)$. For instance, for a path graph $P_4$ with vertices $u_1, v_1, u_2, v_2$ and edges $(u_1, v_1), (v_1, u_2), (u_2, v_2)$, if the vertices are ordered as $u_1, u_2, v_1, v_2$, the greedy coloring uses $3$ colors even though $\chi(P_4) = 2$.

Therefore, statements A and B are correct.