Let k be an integer satisfying k ≥ 2. Let G be a connected graph with...

Let k be an integer satisfying k ≥ 2. Let G be a connected graph with no cycles and k vertices. Prove that G has at least 2 vertices of degree equal to 1.

Solutions

Expert Solution

Colby Messinger answered 11 months ago

Next > < Previous

Related Solutions

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let x be a fixed positive integer. Is it possible to have a graph G with...

Let x be a fixed positive integer. Is it possible to have a graph G with 4x + 1 vertices such that G has a vertex of degree d for all d = 1, 2, ..., 4x + 1? Justify your answer. (Note: The graph G does not need to be simple.)

4. Let n ≥ 8 be an even integer and let k be an integer with...

4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤ k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . , n}. How many such subsets contain at least two even numbers?

Let G be a connected graph which contains two spanning trees T1 and T2 that do...

Let G be a connected graph which contains two spanning trees T1 and T2 that do not share any edges (note that G may contain edges that are in neither trees). Prove that G does not have a bridge. I have a hint for the answer: Show that each edge is in a cycle (2 cases). But I can't figure out the 2 cases. Some help please!

1. Let G be a k-regular bipartite graph. Use Corollary 3.1.13 to prove that G can...

1. Let G be a k-regular bipartite graph. Use Corollary 3.1.13 to prove that G can be decomposed into r-factors iff r divides k.

Let G be connected, and let e be an edge of G. Prove that e is...

Let G be connected, and let e be an edge of G. Prove that e is a bridge if and only if it is in every spanning tree of G.

Let G be an abelian group and K is a subset of G. if K is...

Let G be an abelian group and K is a subset of G. if K is a subgroup of G , show that G is finitely generated if and only if both K and G/K are finitely generated.

Show that there is only one positive integer k such that no graph contains exactly k...

Show that there is only one positive integer k such that no graph contains exactly k spanning trees.

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂...

Let G be a group and K ⊂ G be a normal subgroup. Let H ⊂ G be a subgroup of G such that K ⊂ H Suppose that H is also a normal subgroup of G. (a) Show that H/K ⊂ G/K is a normal subgroup. (b) Show that G/H is isomorphic to (G/K)/(H/K).

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K...

Let G, H, K be groups. Prove that if G ≅ H and H ≅ K then G ≅ K.

Subjects