Question

Find the logical mistakes in these proofs, and explain why the mistakes you've identified cause problems in their arguments. (b)Claim: Suppose that G is a graph on n 3 vertices in which the degree of every vertex is exactly 2. Then G is a cycle Proof. We proceed by induction on n, the number of vertices in G. Our base case is simple: for n - 3, the only graph with 3 vertices in which all vertices have degree 2 is Cs, which is a cycle. is: we assume if G is a graph on n vertices in which every vertex has degree 2 then G is a cycle. Using this, we want to prove our claim for n +1: that is, we want to take any graph G on n+1 vertices in which every vertex has degree 2, and show that G must also be a cycle. Doing this is straightforward! Just take any edge (x, y E G, and perform edge contraction2 on this edge. The result is a graph on n vertices, in which all vertices still have degree 2; therefore, by our inductive assumption, this contracted graph was already a cycle. Now, undo this contraction! This extends our graph to a cycle that's longer by one edge. In particular, it means that our original graph was a cycle, as desired.

Answer 1

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(b) Claim: Suppose that G is a graph on n 3 vertices in which the degree of every vertex is exactly 2. Then G is a cycle. Proof. We proceed by induction on n, the number of vertices in G. Our base case is simple: for n 3, the only graph with 3 vertices in which all vertices have degree 2 is C3, which is a cycle. For our inductive step, we start by assuming that our claim is true for n. That is: we assume if G is a graph on n vertices in which every vertex has degree 2, then G is a cycle Using this, we want to prove our claim for n +1: that is, we want to take any graph G on n1 vertices in which every vertex has degree 2, and show that G must also be a cycle. Doing this is straightforward! Just take any edge r, y E G, and perform edge contraction2 othis edge. The result is a graph on n vertices, in which all vertices still have degree 2; therefore, by our inductive assumption, this contracted graplh was already a cycle Now, undo this contraction! This extends our graph to a cycle that's longer by one edge. In particular, it means that our original graph was a cycle, aธ desired.

EXPLANATION ::-

Performing an edge-contraction can result in a graph in which not all vertices have degree 2. For example the graph that comprises of 2 copies of C_3 (2 triangles).

Then performing edge contraction for 2 vertices belonging to same triangle results in a triangle and a copy of K_2 which is clearly not a cycle.

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