In the study of graph theory, one important concept is that of a Hamiltonian circuit. A Hamiltonian circuit is a significant topic within the field of mathematics, as it relates to the study of paths and cycles in graphs. In this text, we will explore what a Hamiltonian circuit entails and identify which of the provided statements about it is true.
A Hamiltonian circuit is a path in an undirected or directed graph that visits each vertex exactly once and returns to the starting vertex. This means that the circuit forms a closed loop where every vertex in the graph is included, and none is repeated until the circuit concludes.
The focus here is specifically on the vertices visited within the graph rather than all edges, distinguishing a Hamiltonian circuit from other cycle types, such as Eulerian circuits.
Now, let’s assess each of the statements provided to determine which one accurately describes the nature of a Hamiltonian circuit:
This statement is FALSE. A Hamiltonian circuit does not need to traverse every edge exactly once. Instead, its primary focus is on visiting every vertex exactly once. This characteristic would be relevant for an Eulerian cycle, where the requirement is indeed to traverse every edge.
This statement is also FALSE. While a Hamiltonian circuit visits every vertex exactly once, it does not traverse every edge exactly once. The edges used in the course of visiting the vertices are not constrained in this manner; they are merely the connections that facilitate the path taken.
This statement is FALSE. A Hamiltonian circuit, by definition, must return to the starting vertex. If it did not, it would no longer be categorized as a circuit, but as a Hamiltonian path instead.
This statement is TRUE. This accurately encapsulates the definition of a Hamiltonian circuit. It highlights the essential elements of this concept: visiting every vertex in the graph exactly one time and returning to the original starting vertex.
In conclusion, understanding the properties and definitions surrounding Hamiltonian circuits is vital for grasping more complex topics within graph theory. Among the statements presented, the only true assertion is:
D. It visits every vertex exactly once and starts and ends at the same vertex.
This definition is fundamental for distinguishing Hamiltonian circuits from other graph-related paths and cycles, and it serves as a cornerstone in both theoretical and applied mathematics settings. As such, recognizing these distinctions can enhance our comprehension of graph properties and pave the way for deeper explorations into the field.