Question

COMP Discrete Structures: Please answer completely and clearly.

(3).

(5).

x) (4 points) If k is a positive integer, a k-coloring of a graph G is an assignment of one of k possible colors to each of the vertices/edges of G so that adjacent vertices/edges have different colors. Draw pictures of each of the following (a) A 4-coloring of the edges of the Petersen graph. (b) A 3-coloring of the vertices of the Petersen graph. (e) A 2-coloring (d) A 4-coloring of the edges of T2 (see problem 7) of the vertices of K3.4 (a) (4 points) Prove that a graph is bipartite if and only if there is a 2-coloring (see problem 3) of its vertices. (b) (4 points) Prove that if a graph is a tree with at least two vertices, then there is a 2-coloring of its vertices. (Hint: Here are two strategies: (1) use induction on the number of vertices, using the fact that a tree always has a vertez of degree 1, or (2) make the tree rooted, and consider the levels of each verter.) (c) (1 point) Parts (a) and (b) together imply that every tree is bipartite. Show that the converse is false, i.e., draw a bipartite graph that is not a tree.

Answer 1

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