Alternative Proof using the Video

Oct 13, 2025 by miodeya.com 99 views

Is there in every non-complete graph with maximum vertex degree at least 3 a vertex which has two neighbours which are not neighbours to each other?

In graph theory, a branch of mathematics that deals with the study of graphs, a graph is a non-complete graph if it is not a complete graph. A complete graph is a graph where every vertex is connected to every other vertex. A non-complete graph, on the other hand, is a graph where not every vertex is connected to every other vertex. In this article, we will explore the concept of a non-complete graph with a maximum vertex degree of at least 3 and determine if there exists a vertex in such a graph that has two neighbors which are not neighbors to each other.

What is a Maximum Vertex Degree?

The maximum vertex degree of a graph is the maximum number of edges that can be incident on a single vertex. In other words, it is the maximum number of edges that can be connected to a single vertex. For example, in a graph with 5 vertices, if one vertex has 4 edges connected to it, then the maximum vertex degree of that graph is 4. If all vertices have a maximum vertex degree of at least 3, then it means that every vertex in the graph has at least 3 edges connected to it.

Brook's Theorem

Brook's theorem is a fundamental result in graph theory that states that every graph with maximum vertex degree at least 3 is 3-colorable. In other words, it states that every graph with maximum vertex degree at least 3 can be colored with 3 colors such that no two adjacent vertices have the same color. However, the question we are trying to answer is not whether the graph is 3-colorable, but whether there exists a vertex in the graph that has two neighbors which are not neighbors to each other.

Proof of the Existence of a Vertex with Two Non-Neighbor Neighbors

To prove the existence of a vertex with two non-neighbor neighbors in a non-complete graph with maximum vertex degree at least 3, we can use a simple argument. Let's consider a graph G with maximum vertex degree at least 3. Let v be a vertex in G. Since the maximum vertex degree of G is at least 3, v has at least 3 neighbors. Let u and w be two neighbors of v. Since G is not a complete graph, there exists a vertex x that is not a neighbor of v. Now, let's consider the subgraph H induced by the vertices v, u, w, and x. Since v has at least 3 neighbors, H is a subgraph of G with at least 4 vertices. Since the maximum vertex degree of G is at least 3, the maximum vertex degree of H is also at least 3. Now, let's consider the neighbors of u and w in H. Since u and w are neighbors of v, they are not neighbors of each other. Since the maximum vertex degree of H is at least 3, u and w have at least 3 neighbors in H. Let y be a neighbor of u in H that is not a neighbor of w. Then, y is a vertex in H that has two non-neighbor neighbors, namely u and w.

Alternative Proof using the Video

The video mentioned in the additional information presents two different proofs of Brook's theorem. The first proof is the usual, cumbersome argument that seems to be difficult to understand. However, the second proof is a simple and elegant argument that uses the concept of a "star" graph. A star graph is a graph with a central vertex connected to all other vertices. The second proof shows that every graph with maximum vertex degree at least 3 can be transformed into a star graph by adding edges between non-adjacent vertices. This transformation preserves the maximum vertex degree of the graph. Since the star graph has a central vertex connected to all other vertices, it is clear that every vertex in the star graph has two non-neighbor neighbors. Therefore, every graph with maximum vertex degree at least 3 has a vertex with two non-neighbor neighbors.

In conclusion, we have shown that every non-complete graph with maximum vertex degree at least 3 has a vertex that has two neighbors which are not neighbors to each other. This result is a fundamental property of graphs with maximum vertex degree at least 3 and has important implications for graph theory and its applications. The proof of this result is simple and elegant, and it uses the concept of a subgraph and the properties of graphs with maximum vertex degree at least 3.
Is there in every non-complete graph with maximum vertex degree at least 3 a vertex which has two neighbours which are not neighbours to each other? - Q&A

In our previous article, we explored the concept of a non-complete graph with a maximum vertex degree of at least 3 and determined that there exists a vertex in such a graph that has two neighbors which are not neighbors to each other. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the significance of the maximum vertex degree in a graph?

A: The maximum vertex degree of a graph is the maximum number of edges that can be incident on a single vertex. It is an important parameter in graph theory and has implications for the coloring and other properties of the graph.

Q: How does the maximum vertex degree relate to the coloring of a graph?

A: The maximum vertex degree of a graph is related to its coloring. According to Brook's theorem, every graph with maximum vertex degree at least 3 is 3-colorable. This means that every graph with maximum vertex degree at least 3 can be colored with 3 colors such that no two adjacent vertices have the same color.

Q: What is a star graph, and how is it related to the maximum vertex degree?

A: A star graph is a graph with a central vertex connected to all other vertices. The second proof of Brook's theorem uses the concept of a star graph to show that every graph with maximum vertex degree at least 3 can be transformed into a star graph by adding edges between non-adjacent vertices. This transformation preserves the maximum vertex degree of the graph.

Q: How does the existence of a vertex with two non-neighbor neighbors relate to the maximum vertex degree?

A: The existence of a vertex with two non-neighbor neighbors in a graph with maximum vertex degree at least 3 is a fundamental property of such graphs. It has important implications for the coloring and other properties of the graph.

Q: Can you provide an example of a graph with maximum vertex degree at least 3 that has a vertex with two non-neighbor neighbors?

A: Yes, consider a graph with 5 vertices, where one vertex has 4 edges connected to it, and the other vertices have 3 edges connected to them. This graph has a maximum vertex degree of at least 3 and has a vertex with two non-neighbor neighbors.

Q: How does the maximum vertex degree relate to the number of edges in a graph?

A: The maximum vertex degree of a graph is related to the number of edges in the graph. According to the handshaking lemma, the sum of the degrees of all vertices in a graph is equal to twice the number of edges in the graph. Therefore, if the maximum vertex degree of a graph is at least 3, then the graph must have at least 3 edges.

Q: Can you provide a proof of the existence of a vertex with two non-neighbor neighbors in a graph with maximum vertex degree at least 3?

A: Yes, the proof is as follows: Let G be a graph with maximum vertex degree at least 3. Let v be a vertex in G. Since the maximum vertex degree of G is at least 3, v has at least 3 neighbors. Let u and w be two neighbors of v. Since G is not a complete graph, there exists a vertex x that is not a neighbor of v. Now, let's consider the subgraph H induced by the vertices v, u, w, and x. Since v has at least 3 neighbors, H is a subgraph of G with at least 4 vertices. Since the maximum vertex degree of G is at least 3, the maximum vertex degree of H is also at least 3. Now, let's consider the neighbors of u and w in H. Since u and w are neighbors of v, they are not neighbors of each other. Since the maximum vertex degree of H is at least 3, u and w have at least 3 neighbors in H. Let y be a neighbor of u in H that is not a neighbor of w. Then, y is a vertex in H that has two non-neighbor neighbors, namely u and w.

In conclusion, we have answered some frequently asked questions related to the existence of a vertex with two non-neighbor neighbors in a graph with maximum vertex degree at least 3. We have shown that such a vertex exists in every non-complete graph with maximum vertex degree at least 3 and have provided a proof of this result.