Minimal and minimum size latin bitrades of each genus

Lefevre, James, et al. "Minimal and minimum size latin bitrades of each genus." Commentationes Mathematicae Universitatis Carolinae 48.2 (2007): 189-203. <http://eudml.org/doc/250195>.

@article{Lefevre2007,
abstract = {Suppose that $T^\{\circ \}$ and $T^\{\star \}$ are partial latin squares of order $n$, with the property that each row and each column of $T^\{\circ \}$ contains the same set of entries as the corresponding row or column of $T^\{\star \}$. In addition, suppose that each cell in $T^\{\circ \}$ contains an entry if and only if the corresponding cell in $T^\{\star \}$ contains an entry, and these entries (if they exist) are different. Then the pair $T=(T^\{\circ \},T^\{\star \})$ forms a latin bitrade. The size of $T$ is the total number of filled cells in $T^\{\circ \}$ (equivalently $T^\{\star \}$). The latin bitrade is minimal if there is no latin bitrade $(U^\{\circ \},U^\{\otimes \})$ such that $U^\{\circ \}\subseteq T^\{\circ \}$. Drápal (2003) represented latin bitrades in terms of row, column and entry cycles, which he proved formed a coherent digraph. This digraph can be considered as a combinatorial surface, thus associating each latin bitrade with an integer genus, which is a robust structural property of the latin bitrade. For each genus $g\ge 0$, we construct a latin bitrade of smallest possible size, and also a minimal latin bitrade of size $8g+8$.},
author = {Lefevre, James, Donovan, Diane, Cavenagh, Nicholas J., Drápal, Aleš},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {latin trade; bitrade; genus; latin trade; bitrade; genus},
language = {eng},
number = {2},
pages = {189-203},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Minimal and minimum size latin bitrades of each genus},
url = {http://eudml.org/doc/250195},
volume = {48},
year = {2007},
}

TY - JOUR
AU - Lefevre, James
AU - Donovan, Diane
AU - Cavenagh, Nicholas J.
AU - Drápal, Aleš
TI - Minimal and minimum size latin bitrades of each genus
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2007
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 48
IS - 2
SP - 189
EP - 203
AB - Suppose that $T^{\circ }$ and $T^{\star }$ are partial latin squares of order $n$, with the property that each row and each column of $T^{\circ }$ contains the same set of entries as the corresponding row or column of $T^{\star }$. In addition, suppose that each cell in $T^{\circ }$ contains an entry if and only if the corresponding cell in $T^{\star }$ contains an entry, and these entries (if they exist) are different. Then the pair $T=(T^{\circ },T^{\star })$ forms a latin bitrade. The size of $T$ is the total number of filled cells in $T^{\circ }$ (equivalently $T^{\star }$). The latin bitrade is minimal if there is no latin bitrade $(U^{\circ },U^{\otimes })$ such that $U^{\circ }\subseteq T^{\circ }$. Drápal (2003) represented latin bitrades in terms of row, column and entry cycles, which he proved formed a coherent digraph. This digraph can be considered as a combinatorial surface, thus associating each latin bitrade with an integer genus, which is a robust structural property of the latin bitrade. For each genus $g\ge 0$, we construct a latin bitrade of smallest possible size, and also a minimal latin bitrade of size $8g+8$.
LA - eng
KW - latin trade; bitrade; genus; latin trade; bitrade; genus
UR - http://eudml.org/doc/250195
ER -