Question Solved1 Answer 2. Graph coloring Our goal is to color the nodes of a graph such that the endpoints of every edge have different color. Prove that a graph where every node has degree at most k can be colored with k + 1 colors. Hint. We might try induction either on k, the number of colors or on n, the number n of graph nodes. A bit of thinking tells us it's hard to relate in the inductive step if we change the degree of every node in the graph. So consider doing induction on n. The discussion handout gives a real Java class definition for boolean expressions, somewhat similar to the definition of Lisp lists in Section 4.10 (which we aren't doing this semester anyway). Note that there are not separate node and tree objects - a tree can be a single leaf node or a composite node with two subtrees. Writing

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Transcribed Image Text: 2. Graph coloring Our goal is to color the nodes of a graph such that the endpoints of every edge have different color. Prove that a graph where every node has degree at most k can be colored with k + 1 colors. Hint. We might try induction either on k, the number of colors or on n, the number n of graph nodes. A bit of thinking tells us it's hard to relate in the inductive step if we change the degree of every node in the graph. So consider doing induction on n. The discussion handout gives a real Java class definition for boolean expressions, somewhat similar to the definition of Lisp lists in Section 4.10 (which we aren't doing this semester anyway). Note that there are not separate node and tree objects - a tree can be a single leaf node or a composite node with two subtrees. Writing
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Transcribed Image Text: 2. Graph coloring Our goal is to color the nodes of a graph such that the endpoints of every edge have different color. Prove that a graph where every node has degree at most k can be colored with k + 1 colors. Hint. We might try induction either on k, the number of colors or on n, the number n of graph nodes. A bit of thinking tells us it's hard to relate in the inductive step if we change the degree of every node in the graph. So consider doing induction on n. The discussion handout gives a real Java class definition for boolean expressions, somewhat similar to the definition of Lisp lists in Section 4.10 (which we aren't doing this semester anyway). Note that there are not separate node and tree objects - a tree can be a single leaf node or a composite node with two subtrees. Writing
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The statement to prove:- A graph in which every node has degree at most k can be colored with k+1 colors Obviously k >= 0  => k+1 >= 1 [degree cannot be negative]  Proof by induction on number of vertices n:-  Base Case:- when n = 1, then the graph with single vetex can be colored with 1 color. SInce degree of single vertex must be 0 , k+1 >= 1. Hence the statement holds for graph with 1 vertex Induction Hypothesis:- Assume that for all graphs G with n vertices such that degree of each vertex is at most k, G can be colored with at most k+1 colors. Induction step:- Consider a graph G' with n+1 vertices such that degree of any vertex v ... See the full answer