Prove that if G is a simple graph with |V (G)| = n even, where δ(G)...

Prove that if G is a simple graph with |V (G)| = n even, where δ(G) ≥ n 2 + 1, then G has a 3-regular spanning subgraph.

Expert Solution

Colby Messinger answered 2 years ago

Let G be a simple undirected graph with n vertices where n is an even number....

Let G be a simple undirected graph with n vertices where n is an even number. Prove that G contains a triangle if it has at least (n^2 / 4) + 1 edges using mathematical induction.

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has...

Prove or disprove: If G = (V; E) is an undirected graph where every vertex has degree at least 4 and u is in V , then there are at least 64 distinct paths in G that start at u.

Graph Theory Prove: If G is a graph for which deg(u)+deg(v) ≥n for each uv ∈EsubG,...

Graph Theory Prove: If G is a graph for which deg(u)+deg(v) ≥n for each uv ∈EsubG, the G has a Hamiltonian cycle. (with counter examples)

Let G be a simple graph. Prove that the connection relation in G is an equivalence...

Let G be a simple graph. Prove that the connection relation in G is an equivalence relation on V (G)

Problem 2. Consider a graph G = (V,E) where |V|=n. 2(a) What is the total number...

Problem 2. Consider a graph G = (V,E) where |V|=n. 2(a) What is the total number of possible paths of length k ≥ 0 in G from a given starting vertex s and ending vertex t? Hint: a path of length k is a sequence of k + 1 vertices without duplicates. 2(b) What is the total number of possible paths of any length in G from a given starting vertex s and ending vertex t? 2(c) What is the...

Show that any graph with n vertices and δ(G) ≥ n/2 + 1 has a triangle.

Question 1 a) Prove that if u and v are distinct vertices of a graph G,...

Question 1 a) Prove that if u and v are distinct vertices of a graph G, there exists a walk from u to v if and only if there exists a path (a walk with distinct vertices) from u to v. b) Prove that a graph is bipartite if and only if it contains no cycles of odd length. Please write legibly with step by step details. Many thanks!

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3...

A maximal plane graph is a plane graph G = (V, E) with n ≥ 3 vertices such that if we join any two non-adjacent vertices in G, we obtain a non-plane graph. a) Draw a maximal plane graphs on six vertices. b) Show that a maximal plane graph on n points has 3n − 6 edges and 2n − 4 faces. c) A triangulation of an n-gon is a plane graph whose infinite face boundary is a convex n-gon...

Prove or disprove: (a) If G is a graph of order n and size m with...

Prove or disprove: (a) If G is a graph of order n and size m with three cycles, then m ≥ n + 2. (b) There exist exactly two regular trees.

Prove that if n is an integer and n^2 is even the n is even.

Question