<abstract xmlns="http://www.w3.org/1999/xhtml">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 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ausm-2017-0014_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mrow><mo>[</mo><mrow><msub><mtext>f</mtext><mtext>i</mtext></msub></mrow><mo>]</mo></mrow><mrow><mtext>i</mtext><mo>=</mo><mn>1</mn></mrow><mtext>n</mtext></msubsup></math><tex-math>$\left[ {{\rm{f}}_{\rm{i}} } \right]_{{\rm{i}} = 1}^{\rm{n}} $</tex-math></alternatives></inline-formula>
 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.</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 
[fi]i=1n$\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.

Tournaments, oriented graphs and football sequences

Acta Universitatis Sapientiae, Mathematica

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"Tournaments, oriented graphs and football sequences"}

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...

Tournaments, oriented graphs and football sequences

Published Online: Aug 05, 2017

Page range: 213 - 223

Received: Sep 24, 2016

DOI: https://doi.org/10.1515/ausm-2017-0014

Keywords
tournament, oriented graph, scores, football sequence

© 2017 S. Pirzada et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Tournaments, oriented graphs and football sequences

Published Online: Aug 05, 2017

Page range: 213 - 223

Received: Sep 24, 2016

DOI: https://doi.org/10.1515/ausm-2017-0014

Keywordstournament, oriented graph, scores, football sequence

© 2017 S. Pirzada et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
tournament, oriented graph, scores, football sequence