Question

Let \(G=(V, E)\) be a bipartite graph with vertex partition \(V=A \cup B\). Prove that if \(|A| \neq|B|,\) then \(G\) is not Hamiltonian.

Short answer

If \(|A| \neq |B|\), \(G\) cannot have a Hamiltonian cycle.

Step-by-step solution

Understand the problem

We are given a bipartite graph \(G\) with vertex partition \(V = A \cup B\) and we need to show that if \(|A| eq |B|\), then \(G\) cannot have a Hamiltonian cycle.

Define Hamiltonian cycle

A Hamiltonian cycle is a cycle in a graph that visits every vertex exactly once and returns to the starting vertex. For a bipartite graph with vertex sets \(A\) and \(B\), any cycle must alternate between vertices in \(A\) and vertices in \(B\).

Analyze the conditions for a Hamiltonian cycle in a bipartite graph

For a Hamiltonian cycle to exist in a bipartite graph, it must use every vertex, alternating between sets \(A\) and \(B\). This requires the cycle to have an equal number of vertices from both \(A\) and \(B\) at each step around the cycle.

Consider the implication of |A| ≠ |B|

If \(|A| eq |B|\), it is impossible to have an alternating cycle that starts and ends at the same point (i.e., a Hamiltonian cycle), because the cycle would either run out of vertices in \(A\) or \(B\) before completing a loop.

Conclude the proof

Since a Hamiltonian cycle requires starting and ending on the same partition when complete, and we cannot do this if \(|A| eq |B|\) because one side will run out before the other, \(G\) cannot be Hamiltonian.

Key concepts

Hamiltonian Cycle

A Hamiltonian Cycle is an intriguing concept in graph theory. It refers to a path in a graph that visits every vertex exactly once and returns to the starting point. Essentially, it's a route that covers all vertices in the graph without repetition. This cycle is named after the mathematician, Sir William Rowan Hamilton, who studied these paths. In the context of bipartite graphs, a Hamiltonian cycle becomes slightly more intricate due to the vertex partitioning of the graph into two sets, commonly denoted as \(A\) and \(B\).

For such a cycle to occur in a bipartite graph, it must alternate between a vertex in set \(A\) and a vertex in set \(B\), and vice versa. This alternating pattern is crucial, as the cycle must include every vertex in both sets without reusing any. Therefore, the number of vertices in sets \(A\) and \(B\) needs to be the same to maintain the alternate pattern and return to the starting vertex. When these conditions aren't met, forming a complete cycle without missing vertices becomes impossible, explaining the impossibility of a Hamiltonian cycle in such a scenario.

Graph Theory

Graph Theory is a fascinating area of mathematics dealing with graphs, which are abstract representations consisting of vertices (or nodes) connected by edges. It serves as a foundational aspect of modern computational theory, helping to solve complex problems in networks and structures. The core units of a graph—vertices and edges—combine to form paths, cycles, and other constructs.

A special type of graph is known as a bipartite graph, consisting of two disjoint sets of vertices such that every edge connects a vertex from one set to the other. This structure ensures that there are no edges connecting vertices within the same set. Graph theory's applications are widespread, from social network analysis to molecular chemistry, showcasing its versatile nature.

• Variants such as directed graphs, weighted graphs, and more add complexity to the study.
• Concepts like Eulerian paths, spanning trees, and more fall within this broad realm.

Understanding these basic principles can offer a deep insight into a world governed by relationships and connections.

Vertex Partition

Vertex Partition in graph theory refers to the division of a graph's vertices into distinct sets. This partition is often critical in understanding the graph's properties and behavior. An essential form of vertex partition is seen in bipartite graphs, where the vertex set \(V\) splits into two disjoint subsets, \(A\) and \(B\). This setup ensures that edges only stretch from a vertex in one subset to a vertex in the other, not within the same subset.

This systematic arrangement holds significant importance in problems like matching theory, network flow, and scheduling. The partition helps in deducing numerous characteristics, such as the graph’s ability to have a matching or establishing the conditions for the existence of a Hamiltonian cycle.

• Vertex partitions can also generalize into more than two sets, leading to multipartite graphs.
• The role of vertex partitions goes beyond structures by aiding algorithm development and optimizing solutions in computational tasks.

Embarking on the understanding of vertex partitions opens pathways to resolving a variety of both theoretical and practical challenges.