In my understanding this is $K_8$, and the question is how to obtain a colouring $c:E(K_8) \rightarrow \{1,\dots,7\}$ such that $c^{-1}(j)$ is a matching for every $j =1, \dots ,7$.
Because this way each colour would represent the number of the round $($there must be $7$ rounds and each team must play once in each round, but every team needs to play every other team exactly once in the tournament$)$
The condition that $K_8$ is the complete $8$-vertices graph takes care of the "every team plays every other team exactly once" part, and the matching condition on the colouring ensures every team plays exactly one game at each round.
But even if I wrote this problem in graph theory notation I fail to actually obtain the colouring… How can I obtain that?
Best Answer
Given in a tournament there are $8$ teams and let us consider them from $1$ to $8$
Take a regular $n-1$ sided polygon $($Here we choose heptagon for $8$ teams$)$
Draw horizontal stripes as shown below. Then, join the vertex that has been left out to the centre. Each segment represents teams playing each other in the first round.
So, in the first round the teams $(7, 6), (1, 5), (2, 4) \mbox{and} (3, 8) $ play.
Continue rotating the polygon until it returns to its original position.
\begin{array}{c|c|c|c|c|c|c}
\mbox{ Round}&\mbox{ A}&\mbox{B}&\mbox{C}&\mbox{D}\\\hline
I&(7,6)&(1,5)&(2,4)&(3,8)\\
II&(6,5)&(7,4)&(1,3)&(2,8)\\
III&(5,4)&(6,3)&(7,2)&(1,8)\\
IV&(4,3)&(5,2)&(6,1)&(7,8)\\
V&(3,2)&(4,1)&(5,7)&(6,8)\\
VI&(2,1)&(3,7)&(4,6)&(5,8)\\
VII&(1,7)&(2,6)&(3,5)&(4,8)\\
\
\\
\end{array}