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Chapter 2: Problem 52

Prove that the Petersen graph is non-Hamiltonian.

Short Answer

Expert verified
By considering each vertex in the Petersen graph and the paths connected to them, it can be observed no Hamiltonian cycle can exist due to the specific topology. Therefore, the Petersen graph is non-Hamiltonian.

Step by step solution

01

Understanding the Petersen graph

The Petersen graph is a specific type of graph with 10 vertices and 15 edges. This graph is characterized as 3-regular because every one of its vertices has degree 3 (that is, there are three edges coming out of each vertex). It also consists of 5 vertices aligned in a circle (forming a pentagon), and for each vertex in the pentagon, there is a corresponding vertex in the inner star.
02

Approach for proving non-Hamiltonian

To demonstrate that the graph is non-Hamiltonian, it should be shown that no cycle can visit each vertex exactly once. This could be achieved by considering the approach of contradiction. This means, initially assuming that there is a Hamiltonian cycle in the Petersen graph and by arriving at a contradiction.
03

Refute the assumption of a Hamiltonian cycle

Assuming that a Hamiltonian cycle exists in the Petersen graph, there should be a cycle that visits every vertex exactly once. However, the topological structure of the Petersen graph doesn't allow for a Hamiltonian cycle. The vertices of the inner star are connected to three vertices each. As each vertex must be visited exactly once, visiting an inner star vertex would prevent from returning to the outer pentagon on a different path. This leads to a contradiction meaning no Hamiltonian cycle exists.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Theory
At its core, Graph Theory is a field within mathematics and computer science that studies the properties of graphs鈥攁 collection of nodes, or vertices, connected by edges. Graphs are powerful tools used to model relationships in various disciplines ranging from sociology to physics. A graph is formally defined as a set of vertices and a set of edges that connect pairs of vertices. Understanding the structure of a graph, such as the Petersen graph, is crucial in determining its properties, such as whether it's Hamiltonian, which relates to a cycle that visits every vertex exactly once.
Hamiltonian Cycle
A Hamiltonian Cycle in graph theory denotes a loop that travels through each vertex in a graph exactly once and returns to the starting vertex. It's a critical concept for solving routing and network problems. For instance, in the context of the Petersen graph, we look for a cycle that would visit all 10 of its vertices one time. Finding a Hamiltonian cycle can be seen as a version of the classic traveling salesman problem, where the 'salesman' must visit a series of locations exactly once and return to the starting point. Unfortunately, not all graphs are Hamiltonian; in fact, the Petersen graph is an example of a non-Hamiltonian graph.
Proof by Contradiction
The method of Proof by Contradiction is a fundamental logical principle used in mathematics. It begins by assuming the opposite of what you aim to prove. If this assumption leads to a contradiction, you conclude that the assumption must be false and thereby the opposite statement is true. This indirect approach is particularly helpful when a direct proof is challenging. In the exercise, the assumption that the Petersen graph is Hamiltonian leads to a logical inconsistency due to the graph's specific topology, which has a built-in inability to maintain a cyclical pattern that visits each vertex just once.
Regular Graph
A Regular Graph is one where every vertex has the same number of neighboring vertices, known as the degree. In other words, all vertices are uniform in terms of connectivity. The degree of each vertex determines whether a graph is '2-regular', '3-regular', and so on. The Petersen graph is described as being 3-regular, as every one of its vertices is connected to exactly three other vertices. This regularity is a double-edged sword鈥攊t allows certain structural predictions and simplifications but also imposes constraints on the graph's features, particularly impacting the potential for Hamiltonian cycles.

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Most popular questions from this chapter

(i) For which values of \(n\) is \(K_{n}\) Eulerian? (ii) Which complete bipartite graphs are Eulerian? (iii) Which Platonic graphs are Eulerian? (iv) For which values of \(n\) is the wheel \(W_{n}\) Eulerian? (v) For which values of \(k\) is the \(k\)-cube \(Q_{k}\) Eulerian?

Let \(G\) be a Hamiltonian graph and let \(S\) be any set of \(k\) vertices in \(G\). Prove that the graph \(G-S\) has at most \(k\) components.

(i) Let \(G\) be a graph with \(n\) vertices and \(\\{1 / 2(n-1)(n-2)\\}+2\) edges. Use Ore's theorem to prove that \(G\) is Hamiltonian. (ii) Find a non-Hamiltonian graph with \(n\) vertices and \(\left\\{{ }^{1} / 2(n-1)(n-2)\right\\}+1\) edges.

Let \(G\) be a simple graph on \(2 k\) vertices containing no triangles. Prove, by induction on \(k\), that \(G\) has at most \(k^{2}\) edges, and give an example of a graph for which this upper bound is achieved. (This result is often called Tur谩n's extremal theorem.)

Prove that a simple graph and its complement cannot both be disconnected.
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