We say a graph is k-regular if every vertex has degree exactly k. In each of...

We say a graph is k-regular if every vertex has degree exactly k. In each of the following either give a presentation of the graph or show that it does not exist. 1) 3-regular graph on 2018 vertices. 2) 3-regular graph on 2019 vertices.

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Colby Messinger answered 2 years ago

Suppose that a graph G is such that each vertex of G has degree at least...

Suppose that a graph G is such that each vertex of G has degree at least 100. Show that G contains a cycle of length at least 101, i.e., a cycle passing through at least 101 vertices.

Prove that any graph where every vertex has degree at most 3 can be colored with...

Prove that any graph where every vertex has degree at most 3 can be colored with 4 colors.

is there a polyhedron that has 12 hexagonal faces and every vertex of degree 4?

(a) What is the maximum degree of a vertex in a simple graph with n vertices?...

(a) What is the maximum degree of a vertex in a simple graph with n vertices? (b) What is the maximum number of edges in a simple graph of n vertices? (c) Given a natural number n, does there exist a simple graph with n vertices and the maximum number of edges?

find an alternative definition of the degree of a vertex of a graph. please be detailed...

find an alternative definition of the degree of a vertex of a graph. please be detailed as possible thank you

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.

Use the fact that every planar graph with fewer than 12 vertices has a vertex of...

Use the fact that every planar graph with fewer than 12 vertices has a vertex of degree <= 4 to prove that every planar graph with than 12 vertices can be 4-colored.

A tree is a circuit-free connected graph. A leaf is a vertex of degree 1 in...

A tree is a circuit-free connected graph. A leaf is a vertex of degree 1 in a tree. Show that every tree T = (V, E) has the following properties: (a) There is a unique path between every pair of vertices. (b) Adding any edge creates a cycle. (c) Removing any edge disconnects the graph. (d) Every tree with at least two vertices has at least two leaves. (e) | V |=| E | +1.

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.

1.Can you apply Theorem 13.6.12(A Vertex of Degree Five. If GG is a connected planar graph,...

1.Can you apply Theorem 13.6.12(A Vertex of Degree Five. If GG is a connected planar graph, then it has a vertex with degree 5 or less.) to prove that the Three Utilities Puzzle can’t be solved? 2. Prove that if an undirected graph has a subgraph that is a K3K3 it then its chromatic number is at least 3.

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