Question

Prove that if \(D\) is a tournament, then it has a directed Hamiltonian path.

Short answer

Every tournament graph has a directed Hamiltonian path.

Step-by-step solution

Define the Problem

A tournament is a directed graph (digraph) where for every pair of vertices \( (u, v) \), there is exactly one directed edge either from \( u \) to \( v \) or from \( v \) to \( u \). A directed Hamiltonian path is a path that visits each vertex exactly once, following the direction of the edges.

Assume a Vertex Order

To construct a Hamiltonian path, assume an ordering of the vertices of \( D \). Start with an arbitrary vertex and list the remaining vertices in some order. For example, let the vertices be ordered as \( v_1, v_2, \, \dots, \, v_n \).

Construct the Path Iteratively

Starting with \( v_1 \), attempt to extend the path by adding vertices \( v_2, v_3, \, \dots, \, v_n \), at each step ensuring the directed nature of the edges is preserved. If \( v_i \) is connected to \( v_{i+1} \), append \( v_{i+1} \) to the path, else insert \( v_{i+1} \) earlier in the path ensuring the directed edges are valid.

Use Insertion Principally

If the next vertex in the sequence cannot extend the path directly, find an appropriate position in the path such that adding the vertex doesn’t violate the directed path condition. This is possible because, in a tournament, between any two disconnected vertices in the current path, there is a directed edge either way.

Complete the Process

Repeat the insertion process until all vertices are included in the path. The non-existence of cycles in incomplete paths ensures that every vertex can be inserted in a consistent direction, ensuring a directed path covers all vertices.

Key concepts

Tournament Graph

A tournament graph is a special kind of directed graph where every pair of vertices is connected by a single directed edge. This means for any two vertices, either there is a directed edge from the first to the second, or from the second to the first, but never both. Because of this unique property, a tournament graph is always a complete graph in terms of directional edge connections.

In simpler terms, if you imagine this as a competition or a round-robin setup where each player (vertex) plays against every other player exactly once, there is always a winner and a loser (the directed edge pointing from loser to winner).

Key features of tournament graphs include:

• Asymmetric connections between vertices.
• Existence of a path that connects all vertices in a directed manner—just like an itinerary through different match outcomes in a sports tournament.
• No undirected or bidirectional relationships.

Directed Graph

A directed graph, or digraph, is a graph where all the edges have a direction. This means that each edge connects two vertices with a specific direction going from one to the other. Think of it like a one-way street connecting cities, where traffic can only travel in the specified direction.

In practice:

• Every directed edge has a starting vertex (source) and an ending vertex (target).
• The direction is crucial for understanding paths and cycles within the graph.
• Unlike an undirected graph, the direction affects the connection dynamics between vertices.

Directed graphs are used in computer networks, where the data flow must follow specific routes, or in social networks, where the direction could symbolize a one-sided relationship such as a follower or subscriber connection.

Path Construction

Path construction in a graph involves finding a sequence of vertices such that each adjacent pair of vertices in the sequence is connected by an edge following the graph's direction. In the context of a directed Hamiltonian path, the goal is to construct a path that visits each vertex exactly once and respects the direction of each edge.

The construction process begins by:

• Selecting an initial vertex as a starting point.
• Extending the path one vertex at a time, checking that each new vertex maintains the directed structure.
• In a tournament graph, using the properties of vertex connection to ensure a complete path is possible by repositioning vertices when a direct extension isn't possible.

Ultimately, constructing such a path in a tournament graph highlights its inherent structure, always supporting the existence of a directed path visiting all vertices through a clever ordering or insertion approach.

Vertex Ordering

Vertex ordering is the process of arranging the vertices of a graph in a specific sequence. This sequence is crucial in constructing paths, particularly directed Hamiltonian paths, as it determines how vertices will be connected according to the graph's directed edges.

To order vertices effectively in a tournament graph:

• Start with an arbitrary vertex.
• Create an initial list of the remaining vertices in any order.
• Iteratively place each vertex following the directed constraints, either at the end of the current path or by finding a consistent insertion point.

This method of ordering is essential because it enables the finding of a complete path that adheres to the directional nature of the edges in the graph. By carefully ordering vertices, the algorithm ensures that each vertex connects properly to the next one in line, leading to a successful construction of the directed Hamiltonian path.