## Binary 1-factorization:

The**binary 1-factorization**is an 1-factorization (F_{1},F_{2},..., F_{n-1}) of the complete graph K_{n}with even n where the nodes are partitioned into two subsets with n/2 elements.

## Breaks:

A**break**occurs if a team has two consecutive home or away games in two consecutive rounds (then the alternating sequence of home and away games in the home-away pattern of the team is broken).## Canonical 1-factorization:

The**canonical 1-factorization**is an 1-factorization (F_{1},F_{2},..., F_{n-1}) of the complete graph K_{n}with even n.

The 1-factor F_{t}(t=1,...,n-1) contains the edge [n,t] and all edges [(t+k) mod (n-1), (t-k) mod (n-1)] for k=1,...,n/2-1 (where the numbers 1,...,n-1 have to be taken in the mod-expressions).

For odd n after introducing a dummy node n+1 the construction for an even number of nodes can be applied.## Carry-over effect:

Team i gives a so-called**carry-over effect**to team j if some other team k plays first against i in some round t and immediatly after that against team j in round t+1, where the rounds are considered cyclically (i.e. the first round 1 follows the last round n-1).

If several teams play consecutively against i and j and i is a very strong or weak team, team j may be handicapped or favoured compared with other teams. In an `ideal' schedule with respect to balanced carry-over effects no teams i,j,k,l should exist such that teams k and l both play against team j immediately after playing against team i. The best balanced schedules known so far are based on the algebraic structure of a**starter**.## Complementary teams:

In sports tournaments with home-away assignments**complementary teams**are teams with complementary home-away patterns (i.e. one team plays at home if and only if the other team plays away).## Complete graph:

The**complete graph**K_{n}is a graph consisting of n nodes and n(n-1)/2 edges between all pair of nodes.## Directed canonical 1-factorization:

The**directed canonical 1-factorization**is the directed (oriented) version of the canonical 1-factorization (F_{1},F_{2},..., F_{n-1}) of the complete graph K_{n}with even n.

The 1-factor F_{t}(t=1,...,n-1) contains the edge [n,t] and all edges [(t+k) mod (n-1), (t-k) mod (n-1)] for k=1,...,n/2-1 (where the numbers 1,...,n-1 have to be taken in the mod-expressions). The edge [n,t] is oriented into (t,n) if t is odd, and (n,t) if t is even.

The edge [(t+k) mod (n-1), (t-k) mod (n-1)] for k=1,...,n/2-1 is oriented into the arc ((t+k) mod (n-1), (t-k) mod (n-1)) when k is odd and the other way round, when k is even.

For odd n after introducing a dummy node n+1 the construction for an even number of nodes can be applied.

Schedules constructed from the directed canonical 1-factorization have a minimum number of breaks: n-2 for even n and 0 for odd n.## Double Round Robin Tournament (DRRT):

In a**double round robin tournament**(DRRT) each team plays against each other team exactly twice. The tournament is divided into two half series, where in each half series a SRRT is scheduled (with different home teams). A DRRT with n teams may be scheduled as a mirrored DRRT with at least 3(n-2) breaks or according to the English System with 2(n-2) breaks.## Edge coloring:

An**edge coloring**of a graph G=(V,E) is an assignment of colors to all edges from the set E such that edges with a common node receive different colors.## English system:

The**English system**is used for a DRRT when the number of breaks is to be minimized. The first round of the second half series is the same as the last round of the first half series, afterwards the rounds are scheduled in the same order as in the first half series. This system ensures that for n teams only 2(n-2) breaks are needed.## Graph:

An (undirected)**graph**G=(V,E) consists of a finite set V of nodes (vertices) i=1,..,|V| and a set E of (undirected) edges, where each edge [i,j] is an unordered pair of different elements i,j from V. A**directed graph**G=(V,E) consists of a finite set V of nodes (vertices) i=1,..,|V| and a set E of (directed) arcs, where each arc (i,j) is an ordered pair of different elements i,j from V.## Graph model:

A single round robin tournament SRRT for n teams can be represented by the complete graph K_{n}. While the nodes i=1,...,n represent the teams, the edge [i,j] models the game between teams i and j. A schedule corresponds to an edge coloring of the K_{n}with n-1 colors, i.e. an 1-factorization (F_{1},F_{2},..., F_{n-1}), where the 1-factor F_{t}consists of n/2 non-adjacent edges corresponding to the games scheduled in round t.

If additionally home-away patterns have to be calculated, the edge [i,j] is oriented into the arc (i,j) for a home game of team j and into the arc (j,i) for a home game of team i.## Groups:

In tournaments with**groups**the n teams are partitioned into g groups (e.g. according to their strengths) where all groups contain n/g teams. The objective is to find group-changing or even group-balanced schedules.## Group-balanced schedule:

In**group-balanced schedules**for tournaments with g groups no team plays more than once against teams of the same group within g consecutive games. A group-balanced schedule exists if and only if n is odd or n and n/g are both even. A weaker form are group-changing schedules.## Group-changing schedule:

In**group-changing schedules**for tournaments with g groups no team plays two consecutive games against teams of the same group. More restricted are group-balanced schedules.## Home-away pattern (HAP):

A**home-away pattern**for a team states for every round whether the team plays at home or away. Usually, home-away patterns with alternating sequences of home and away games (i.e. without breaks) are prefered.## Mirrored Double Round Robin Tournament:

A**mirrored double round robin tournament**is a DRRT where the rounds of the second half series are scheduled in the same order as in the first half series. For n teams such schedules have at least 3(n-2) breaks.## Oberwolfach problem:

In the Oberwolfach problem (cf. e.g. Colbourn & Dinitz (2006)) the complete graph K_{n}with odd n and integers l_{1},...,l_{t}>=3 with l_{1}+...+l_{t}=n are given. The question is whether the edges of K_{n}can be decomposed into (n-1)/2 2-factors where each 2-factor consists of t cycles with lengths l_{1},...,l_{t}.## One-factorization (1-factorization):

An**1-factorization**of a graph G=(V,E) is a partitioning of the edge set E into edge-disjoint 1-factors (F_{1},F_{2},..., F_{f}) with f=2|E|/|V|, where each 1-factor F_{k}is a subset of edges such that every node is incident to exactly one edge in F_{k}.## Opponent schedule:

In an**opponent schedule**the rows are indexed by teams, the columns by rounds. The entry*opp*[i][t] contains the team j against which i plays in round t. In sports tournaments with home-away assignments the entry*opp*[i][t] is j if the game is a home game for team i and -j if the game is a home game for team j. Other representations of schedules are slot schedules or table schedules.## Rounds:

In sports tournaments**rounds**are the days when games are played. If the number n of teams is even, usually n-1 rounds are used; if n is odd, schedules contain n rounds.## Single Round Robin Tournament (SRRT):

In a**single round robin tournament**(SRRT) each team plays against each other team exactly once. If the number n of teams is even, the schedules contain n-1 rounds and each team plays exactly one game in each round. If n is odd, the schedules contain n rounds and in each round one team has a bye (i.e. it does not play). For even n schedules have at least n-2 breaks, for odd n schedules without any breaks exist.## Slot schedule:

In a**slot schedule**the rows and columns are indexed by teams. The entry*slot*[i][j] contains the round t in which teams i and j play against each other. In sports tournaments with home-away assignments the entry*slot*[i][j] is t if the game is a home game for team i and -t if the game is a home game for team j. Other representations of schedules are opponent schedules or table schedules.## Starter:

The pairs {x_{1}, y_{1}},...,{x_{n-1}, y_{n-1}} form a**starter**in the additive group (Z_{n-1},+) if

* the numbers x_{1},...,x_{n-1}, y_{1},...,y_{n-1}are precisely all elements of Z_{n-1}without 0,

* the differences (x_{1}-y_{1}),...,(x_{n-1}-y_{n-1}), (y_{1}-x_{1}),...,(y_{n-1}-x_{n-1}) are also precisely all the nonzero elements of Z_{n-1}.

Any starter along with the pair {0,infinity} gives the first round of a cyclic schedule in which the teams x_{i}have to play against y_{i}for i =1,...,n-1. Starters are used for schedules balancing carry-over effects.## Table schedule:

In a**table schedule**columns are indexed by rounds. Column t contains all games i-j which are scheduled in round t. In sports tournaments with home-away assignments the entry i-j means a home game for team i, the entry j-i means a home game for team j. Other representations of schedules are opponent schedules or slot schedules.## Teams:

In sports tournaments**teams**(e.g. in soccer, icehockey) or individual players (e.g. in tennis, tabletennis) play games against each other. In this web application all teams are identified by the numbers 1,...,n.## Two-factorization (2-factorization):

A**2-factorization**of a graph G=(V,E) is a partitioning of the edge set E into edge-disjoint 2-factors (F_{1},F_{2},..., F_{f}) with f=4|E|/|V|, where each 2-factor F_{k}is a subset of edges such that every node is incident to exactly two edges in F_{k}.