Question

Let \(G=(V, E)\) be a graph. Prove that if the minimum degree of \(G\) is greater than \((|V|-1) / 2,\) then \(G\) is connected.

Short answer

If the minimum degree is greater than \( (|V|-1) / 2 \), then the graph is connected by contradiction.

Step-by-step solution

Define the Graph and Minimum Degree

Let the graph be represented as \( G = (V, E) \), where \( V \) is the set of vertices and \( E \) is the set of edges. The minimum degree \( \delta(G) \) is defined as the smallest degree of any vertex in the graph. We are given that \( \delta(G) > \frac{|V|-1}{2} \).

Consider the Contradiction

Assume, for contradiction, that the graph \( G \) is not connected. This means \( G \) must consist of at least two disjoint subsets of vertices, say \( V_1 \) and \( V_2 \), such that \( V_1 \cup V_2 = V \) and there are no edges between \( V_1 \) and \( V_2 \).

Apply Minimum Degree Condition

Since there are no edges between \( V_1 \) and \( V_2 \), the degree of each vertex in \( V_1 \) is at most \( |V_1| - 1 \) and similarly for \( V_2 \). However, since the minimum degree \( \delta(G) > \frac{|V|-1}{2} \), every vertex in \( G \) would require degree greater than half of the total vertices minus 1, which makes it impossible to have the partitions described.

Conclude The Graph is Connected

Since having two disjoint subsets without connections contradicts the minimum degree condition, the assumption that \( G \) is not connected must be false. Therefore, \( G \) must be connected.

Key concepts

Minimum Degree

The minimum degree of a graph is an important concept in graph theory. It represents the smallest degree of any vertex in the graph, which essentially counts how many edges are connected to a particular vertex. This concept helps us understand the connectivity of a graph; the higher the minimum degree, the more connected the graph tends to be.
In this context, we are particularly concerned with situations where the minimum degree exceeds half of one less than the total number of vertices in the graph, i.e., \( \delta(G) > \frac{|V|-1}{2} \). This condition is crucial because it sets a threshold for connectivity, suggesting that each vertex is connected to enough others to help form a single interconnected whole.

Disjoint Subsets

When analyzing graphs, the idea of disjoint subsets helps us understand connectivity. If a graph is not connected, it means it can be split into two or more parts that have no interconnecting edges.
These parts are known as disjoint subsets of vertices. In graph theory, disjoint subsets are like islands in a sea of possible connections, where no bridges (edges) exist between them.
For a graph to defy being connected, there need to be these two disjoint vertex sets such as \(V_1\) and \(V_2\), with no edges running between them, which helps in forming a basis for contradiction in proofs.

Contradiction Method

The contradiction method is a common modern approach for proving properties in mathematics, especially in graph theory. It involves assuming the opposite of what you want to prove, establishing implications of this assumption, and then showing that these lead to a contradiction.
In this exercise, the assumption starts with the graph being not connected, implying that it should be possible to divide it into disconnected disjoint subsets. However, the constraints given by the minimum degree lead to the conclusion that at least one connection between these subsets must exist.
Since this contradicts the original assumption of disconnection, we conclude that the graph must indeed be connected. The contradiction method effectively pinpoints impossibility, reinforcing the feared outcome as inevitably true.

Graph Theory

Graph theory is a fascinating field of mathematics dealing with the study of graphs, which are collections of nodes (vertices) linked by lines (edges). It's the backbone of networking, computing, and solving complex logistical problems, offering a blueprint for unwrapping how systems interconnect.
Key components of graph theory include vertices, edges, and paths. Understanding these elements helps analyze networks in terms of their connectivity, cycle presence, and more.
In solving graph-related problems such as the one given, concepts like minimum degree can demonstrate how interconnected a graph is, typically ensuring that networks remain uninterrupted and successfully navigate through various connected components. Graph theory takes theoretical and real-world problems, transforming them through calculation and structure into deducible states of connectivity and flow.