Question

Task 4 (2 pt) Theorem. Let G be a graph with following properties: (1). G is a cycle-free. (2). G has n vertices, $n \ge 1$. (3). G has $(n-1)$ edges. Use mathematical induction by number of vertices n to prove that, that is, G is connected. (Hint. Handshaking theorem: $2m = \sum_{v \in V} deg(v)$ can be useful for inductive step of a proof, here m is number of vertices in G, deg(v) - degree of vertex v, V - set of all vertices of G.)

          Task 4 (2 pt)
Theorem. Let G be a graph with following properties:
(1). G is a cycle-free.
(2). G has n vertices, $n \ge 1$.
(3). G has $(n-1)$ edges.
Use mathematical induction by number of vertices n to prove that, that is, G is connected.
(Hint. Handshaking theorem: $2m = \sum_{v \in V} deg(v)$ can be useful for inductive step of a proof, here m is number of
vertices in G, deg(v) - degree of vertex v, V - set of all vertices of G.)

Task 4 (2 pt)
Theorem. Let G be a graph with following properties:
(1). G is a cycle-free.
(2). G has n vertices, n ≥ 1.
(3). G has (n-1) edges.
Use mathematical induction by number of vertices n to prove that, that is, G is connected.
(Hint. Handshaking theorem: 2m = ∑v ∈ V deg(v) can be useful for inductive step of a proof, here m is number of
vertices in G, deg(v) - degree of vertex v, V - set of all vertices of G.)

Added by Eric S.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Emily Anderson

Step 1

Inductive Hypothesis: Assume that for any graph G with k vertices (k ≥ 1) and k-1 edges, G is connected and cycle-free. Inductive Step: Now, consider a graph G' with n+1 vertices and n edges. We want to show that G' is connected and cycle-free. By the Show more…

Show all steps

Thanks for your feedback!

Task 4 (2 pt) Theorem: Let G be a graph with the following properties: 1. G is cycle-free. 2. G has n vertices, denoted as V. 3. G has (n-1) edges. Use mathematical induction by the number of vertices, n, to prove that G is connected. (Hint: The Handshaking theorem: Î£deg(V) can be useful for the inductive step of the proof. Here, Î£ is the sum of the degrees of all vertices in G.)