# Tournament Graphs

Tournament graphs are used in discrete mathematics to represent a winning vertex in a graph. A tournament is a complete graph in which every pair of vertices are connected by a directed edge. These types of graphs are referred to as tournaments because each of the $$n$$ players competes against the other $$n-1$$ players where ties are not allowed and the winner can be represented on a graph. These graphs are created by assigning every player a vertex and if player $$1$$ beats player $$2$$, then a directed edge can be drawn with the arrow pointing from $$1$$ to $$2.$$ Tournaments graphs create Hamiltonian paths that go through each vertex. The Hamiltonian path theorem states that for every tournament there is a Hamiltonian path for $$n\ge1,$$ for any tournament consisting of $$n$$ vertices in which there is always a sequence of vertices $$v_1,v_2,...,v_n$$ such that $$v_1\rightarrow v_2\rightarrow...\rightarrow v_n.$$
For example,