Final answer:
In the category of complete graphs, it is guaranteed that every graph has a Hamilton circuit, as each vertex is connected to every other vertex allowing for a closed path visiting each vertex exactly once. The correct answer is A.
Step-by-step explanation:
The category in which it is guaranteed that every graph has a Hamilton circuit is complete graphs. A Hamilton circuit is a closed path that visits each vertex in the graph exactly once and returns to the starting vertex.
Why Complete Graphs Have a Hamilton Circuit
Complete graphs, denoted as Kn, where n is the number of vertices, have the distinct property that every pair of distinct vertices is connected by a unique edge. This means in a complete graph with n vertices, each vertex has an edge to every other vertex, making it trivial to construct a Hamilton circuit. You simply start at any vertex, follow through each vertex once, and return to the starting vertex to form a circuit. Such a guarantee does not exist in bipartite graphs, tree graphs, or directed graphs for various reasons.
- Bipartite graphs can only form a Hamilton circuit if both subsets of vertices have the same number of vertices, which isn't always the case.
- Tree graphs, being acyclic, have no path that can return to the starting point without revisiting the same node more than once, thus they cannot have a Hamilton circuit.
- Directed graphs (digraphs) may have directed edges that make it infeasible to find a circuit that traverses all vertices exactly once and returns to the starting point.
In summary, complete graphs definitely have a Hamilton circuit due to the full interconnection of vertices, while the other structures may not.