Directed-Graph Modeling of Round-Robin Tournament Outcomes (Beat Relation)

In a round-robin tournament, each pair of teams plays exactly once. We can model the “beat” outcome using a Directed Graph. Let each team be a vertex, and draw a directed edge from the team that wins to the team that loses (e.g., an edge $T \rightarrow B$ means Tigers beat Blue Jays).

This creates a Tournament Graph where for every two distinct teams $u$ and $v$, exactly one of $u \rightarrow v$ or $v \rightarrow u$ is present.

Key terms you’ll use:

• Vertex
• Directed Edge
• Out-degree
• In-degree

Translate the given results into edges

We are told these outcomes:

• Tigers beat Blue Jays
• Tigers beat Cardinals
• Tigers beat Orioles
• Blue Jays beat Cardinals
• Blue Jays beat Orioles
• Cardinals beat Orioles

Using the rule: “winner $\rightarrow$ loser,” the vertex set is $V=\{\text{Tigers},\text{Blue Jays},\text{Cardinals},\text{Orioles}\}.$

And the directed edge set is $E=\{(\text{Tigers}\rightarrow \text{Blue Jays}),(\text{Tigers}\rightarrow \text{Cardinals}),(\text{Tigers}\rightarrow \text{Orioles}),(\text{Blue Jays}\rightarrow \text{Cardinals}),(\text{Blue Jays}\rightarrow \text{Orioles}),(\text{Cardinals}\rightarrow \text{Orioles})\}.$

This is a Complete Oriented Graph with a consistent win/loss direction for each pair.

Edge list (explicit directed graph model)

The directed graph is:

• Tigers $\rightarrow$ Blue Jays
• Tigers $\rightarrow$ Cardinals
• Tigers $\rightarrow$ Orioles
• Blue Jays $\rightarrow$ Cardinals
• Blue Jays $\rightarrow$ Orioles
• Cardinals $\rightarrow$ Orioles

We can also show the edge set as pairs in a compact form:

where $T=\text{Tigers}$, $B=\text{Blue Jays}$, $C=\text{Cardinals}$, $O=\text{Orioles}$.