Tournaments, oriented graphs and football sequences

Abstract

Abstract Consider the result of a soccer league competition where n teams play each other exactly once. A team gets three points for each win and one point for each draw. The total score obtained by each team vi is called the f-score of vi and is denoted by fi. The sequences of all f-scores [ f i ] i = 1 n $\left[ {{\rm{f}}_{\rm{i}} } \right]_{{\rm{i}} = 1}^{\rm{n}} $ arranged in non-decreasing order is called the f-score sequence of the competition. We raise the following problem: Which sequences of non-negative integers in non-decreasing order is a football sequence, that is the outcome of a soccer league competition. We model such a competition by an oriented graph with teams represented by vertices in which the teams play each other once, with an arc from team u to team v if and only if u defeats v. We obtain some necessary conditions for football sequences and some characterizations under restrictions.