B-1 Graph coloring Given an undirected graph G (V. E), a k-coloring of G is a function c : V → {0, 1, . . . ,k-1} such that c(u)≠c(v) for every edge (u, v) ∈ E. In other words, the numbers 0.1,... k-1 represent the k colors, and adjacent vertices different colors. must have
c. Let d be the maximum degree of any vertex in a graph G. Prove that we can color G with d +1 colors.
We use induction on the number of vertices in the graph, which we denote by n. Let P(n) be the proposition that an n-vertex graph with maximum degree at most d is (d + 1)-colorable.
Base case (n = 1):
Inductive step:
Solution:-------------------
We do induction on the number of nodes.
If n ≤ d + 1, this is trivial.
Suppose the result holds for all graphs with ≤ n vertices.
Then given a graph G on n + 1 vertices and maximum degree d, remove
some vertex v to obtain G".
G" has n vertices, and maximum degree at most d, and thus has a d +
1 coloring by our hypothesis. Now simply assign v some color that
is not used by its neighbors (such a color exists as deg(v) ≤
d).
B-1 Graph coloring Given an undirected graph G (V. E), a k-coloring of G is a function c : V → {0, 1, . . . ,k-1}
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