Question

Let \(G=(V, E)\) be a graph. Prove that if for any two nonadjacent vertices \(v\) and \(w\) of \(G\) we have \(\operatorname{deg}(v)+\operatorname{deg}(w) \geq|V|,\) then \(G\) is Hamiltonian.

Short answer

The graph \(G\) is Hamiltonian due to the ensured degree condition.

Step-by-step solution

Understanding the Problem

We are given a graph \(G = (V, E)\) and need to prove that if for any two nonadjacent vertices \(v\) and \(w\) the sum of their degrees is at least the number of vertices \(|V|\), then the graph is Hamiltonian. A Hamiltonian graph is one that contains a Hamiltonian cycle, a cycle passing through each vertex exactly once.

Applying Dirac's Theorem

Dirac's Theorem states that a graph with \(n\) vertices \((n \geq 3)\) is Hamiltonian if every vertex has degree at least \(n/2\). We need to relate the given condition to this theorem or deduce a similar property ensuring a Hamiltonian cycle.

Analyzing Given Condition

For any two nonadjacent vertices \(v\) and \(w\), we have \(\operatorname{deg}(v) + \operatorname{deg}(w) \geq |V|\). This implies each vertex has a certain distribution of edges such that even their unconnected counterparts have significant combined degrees, suggesting dense connectivity.

Using Pósa's Condition on Hamiltonicity

An alternative approach leverages Pósa's condition: if for every subset \(S\) of \(V\), the number of components in the subgraph induced by \(V-S\) is at most \(|S|\), then \(G\) is Hamiltonian. Our condition indicates that removing any subset \(S\) leaves the remaining vertices sufficiently interconnected.

Formulating the Proof

Since for any two nonadjacent vertices \(v\) and \(w\), \(\operatorname{deg}(v) + \operatorname{deg}(w) \geq |V|\), assume \(G\) is not Hamiltonian. Use this to derive a contradiction: the construction of cycles will be hindered in non-Hamiltonian cases only by lacking deeper connectivity, which our condition refutes by ensuring substantial connectivity.

Conclusion and Verification

Since the condition provided satisfies conditions for Hamiltonicity either through Dirac's Theorem implications or by Pósa's connectivity criteria, we conclude the graph \(G\) must be Hamiltonian.

Key concepts

Dirac's Theorem

Dirac's Theorem is a critical result in graph theory, providing sufficient conditions for a graph to be Hamiltonian. A Hamiltonian graph possesses a Hamiltonian cycle, meaning a cycle that visits every vertex exactly once and returns to the starting vertex. However, not all graphs are naturally Hamiltonian, and it can be challenging to identify which are. Here, Dirac's Theorem aids us by offering a simple criterion. In essence, Dirac's Theorem states that if a graph has at least three vertices and every vertex has a degree of at least half the total number of vertices (\(\frac{n}{2}\), where \(n\) is the number of vertices in the graph), then the graph must contain a Hamiltonian cycle. This provides a straightforward test to determine Hamiltonicity without explicitly constructing the cycle. The rationale is that a higher degree signifies greater connectivity and thus more potential paths to form a cycle.Thus, Dirac's Theorem is instrumental in graph theory for quickly assessing the Hamiltonian nature of a graph, supporting both theoretical proofs and practical applications.

Pósa's Condition

Pósa's Condition offers another approach for determining whether a given graph is Hamiltonian, extending beyond the criteria provided by Dirac. Specifically, while Dirac's Theorem focuses on individual vertex degrees, Pósa's Condition analyzes entire subsets of vertices within the graph.This condition states that for any subset \(S\) of vertices, the number of connected components in the subgraph induced by removing \(S\) (denoted \(V - S\)) should not exceed the size of \(S\) for the graph to be Hamiltonian. Essentially, it implies that removing any group of vertices leaves the remaining parts of the graph sufficiently connected to form a cycle.By exploring all possible subsets of vertices and checking this connectivity property, Pósa's Condition gives a powerful tool to verify Hamiltonicity. It may require more computation than Dirac's Theorem but can often assess graphs that the latter may overlook, offering more nuanced insight into graph structures and their potential Hamiltonian cycles.

Graph Theory

Graph theory is a vast area of mathematics focusing on the study of graphs, which are structures used to model pairwise relations between objects. A graph consists of vertices (or nodes) connected by edges. Graphs can represent various real-world systems, from social networks to computer networks, making them a crucial subject for understanding interconnected structures. Key concepts in graph theory include trees, paths, cycles, connectivity, and coloring, among many others. Specifically, a Hamiltonian graph, as discussed, relates to cycles encompassing every vertex exactly once. The existence of such cycles can be complex to determine, sparking interest in conditions like Dirac's Theorem and Pósa's Condition. Graph theory empowers us to solve real-life problems using abstract reasoning, and its methods and results are pivotal in computer science, logistics, and networking. As theories evolve, they contribute significantly to our ability to model and analyze systems efficiently, showcasing the applicability and relevance of graph theory in numerous fields.