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| | Greetings. The writer's title is Phebe and she feels comfy when people use the complete name. Body developing is 1 of the issues I love most. She is a librarian but she's usually needed her own company. Years in the past we moved to North Dakota and I love every working day living here.<br><br>My web page - [http://www.adosphere.com/poyocum www.adosphere.com] |
| | name = Rook's graph
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| | image = [[Image:Rook's graph.svg|180px]]
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| | image_caption = 8x8 Rook's graph
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| | vertices = ''nm''
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| | edges = ''nm''(''n''+''m'')/2 - ''nm''
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| | diameter = 2
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| | chromatic_number = max(''n'',''m'')
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| | chromatic_index =
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| | girth = 3 (if max(''n'',''m'') ≥ 3)
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| | properties = [[regular graph|regular]],<br/>[[vertex-transitive graph|vertex-transitive]],<br/>[[perfect graph|perfect]]<br/>[[well-covered graph|well-covered]]
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| }}
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| In [[graph theory]], a '''rook's graph''' is a graph that represents all legal moves of the [[Rook (chess)|rook]] [[chess]] [[Chess piece|piece]] on a [[chessboard]]: each vertex represents a square on a chessboard and each edge represents a legal move. Rook's graphs are highly symmetric [[perfect graph]]s; they may be characterized in terms of the number of triangles each edge belongs to and by the existence of a 4-cycle connecting each nonadjacent pair of vertices.
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| ==Definitions==
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| An ''n'' × ''m'' rook's graph represents the moves of a rook on an ''n'' × ''m'' chessboard.
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| Its vertices may be given coordinates (''x'',''y''), where 1 ≤ ''x'' ≤ ''n'' and 1 ≤ ''y'' ≤ ''m''. Two vertices (''x''<sub>1</sub>,''y''<sub>1</sub>) and (''x''<sub>2</sub>,''y''<sub>2</sub>) are adjacent if and only if either ''x''<sub>1</sub> = ''x''<sub>2</sub> or ''y''<sub>1</sub> = ''y''<sub>2</sub>; that is, if they belong to the same rank or the same file of the chessboard.
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| For an ''n'' × ''m'' rook's graph the total number of vertices is simply ''nm''. For an ''n'' × ''n'' rook's graph the total number of vertices is simply <math>n^2</math> and the total number of edges is <math>n^3 - n^2</math>; in this case the graph is also known as a two-dimensional [[Hamming graph]] or a [[Latin square]] graph.
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| The rook's graph for an ''n'' × ''m'' chessboard may also be defined as the [[Cartesian product of graphs|Cartesian product]] of two [[complete graph]]s ''K''<sub>''n''</sub> <math>\square</math> ''K''<sub>''m''</sub>. The [[complement graph]] of a 2 × ''n'' rook's graph is a [[crown graph]]. | |
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| ==Symmetry==
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| Rook's graphs are [[Vertex-transitive graph|vertex-transitive]] and (''n'' + ''m'' − 2)-[[Regular graph|regular]]; they are the only regular graphs formed from the moves of standard chess pieces in this way (Elkies). When ''m'' ≠ ''n'', the symmetries of the rook's graph are formed by independently permuting the rows and columns of the graph. When ''n'' = ''m'' the graph has additional symmetries that swap the rows and columns; the rook's graph for a square chessboard is [[symmetric]].
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| Any two vertices in a rook's graph are either at distance one or two from each other, according to whether they are adjacent or nonadjacent respectively. Any two nonadjacent vertices may be transformed into any other two nonadjacent vertices by a symmetry of the graph. When the rook's graph is not square, the pairs of adjacent vertices fall into two [[orbit (group theory)|orbits]] of the symmetry group according to whether they are adjacent horizontally or vertically, but when the graph is square any two adjacent vertices may also be mapped into each other by a symmetry and the graph is therefore [[distance-transitive graph|distance-transitive]].
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| When ''m'' and ''n'' are [[relatively prime]], the symmetry group ''S<sub>m</sub>''×''S<sub>n</sub>'' of the rook's graph contains as a subgroup the [[cyclic group]] ''C<sub>mn</sub>'' = ''C<sub>m</sub>''×''C<sub>n</sub>'' that [[group action|acts]] by cyclically permuting the ''mn'' vertices; therefore, in this case, the rook's graph is a [[circulant graph]].
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| ==Perfection==
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| [[Image:Paley9-perfect.svg|thumb|left|The 3×3 rook's graph, colored with three colors and showing a clique of three vertices. In this graph and each of its induced subgraphs the chromatic number equals the clique number, so it is a perfect graph.]]
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| A rook's graph can also be viewed as the [[line graph]] of a [[complete bipartite graph]] ''K''<sub>''n'',''m''</sub> — that is, it has one vertex for each edge of ''K''<sub>''n'',''m''</sub>, and two vertices of the rook's graph are adjacent if and only if the corresponding edges of the complete bipartite graph share a common endpoint. In this view, an edge in the complete bipartite graph from the ''i''th vertex on one side of the bipartition to the ''j''th vertex on the other side corresponds to a chessboard square with coordinates (''i'',''j'').
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| Any [[bipartite graph]] is a [[Glossary of graph theory#Subgraphs|subgraph]] of a complete bipartite graph, and correspondingly any line graph of a bipartite graph is an [[induced subgraph]] of a rook's graph. The line graphs of bipartite graphs are [[perfect graph|perfect]]: in them, and in any of their induced subgraphs, the number of colors needed in any [[graph coloring|vertex coloring]] is the same as the number of vertices in the largest [[clique (graph theory)|complete subgraph]]. Line graphs of bipartite graphs form an important family of perfect graphs: they are one of a small number of families used by {{harvtxt|Chudnovsky|Robertson|Seymour|Thomas|2006}} to characterize the perfect graphs and to show that every graph with no odd hole and no odd antihole is perfect. In particular, rook's graphs are themselves perfect.
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| Because a rook's graph is perfect, the number of colors needed in any coloring of the graph is just the size of its largest clique. The cliques of a rook's graph are the subsets of its rows and columns, and the largest of these have size max(''m'',''n''), so this is also the chromatic number of the graph. An ''n''-coloring of an ''n''×''n'' rook's graph may be interpreted as a [[Latin square]]: it describes a way of filling the rows and columns of an ''n''×''n'' grid with ''n'' different values in such a way that the same value does not appear twice in any row or column.
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| {{Chess diagram|=
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| | tright
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| 8 |__|__|__|rl|__|__|__|__|=
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| 7 |__|__|__|__|__|__|rl|__|=
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| 6 |__|__|rl|__|__|__|__|__|=
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| 5 |rl|__|__|__|__|__|__|__|=
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| 4 |__|rl|__|__|__|__|__|__|=
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| 3 |__|__|__|__|__|__|__|rl|=
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| 2 |__|__|__|__|rl|__|__|__|=
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| 1 |__|__|__|__|__|rl|__|__|=
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| |A non-attacking placement of eight rooks on a chessboard, forming a maximum independent set in the corresponding rook's graph
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| }}
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| An [[Independent set (graph theory)|independent set]] in a rook's graph is a set of vertices, no two of which belong to the same row or column of the graph; in chess terms, it corresponds to a placement of rooks no two of which attack each other. Perfect graphs may also be described as the graphs in which, in every induced subgraph, the size of the largest independent set is equal to the number of cliques in a partition of the graph's vertices into a minimum number of cliques. In a rook's graph, the sets of rows or the sets of columns (whichever has fewer sets) form such an optimal partition. The size of the largest independent set in the graph is therefore min(''m'',''n''). Every color class in every optimal coloring of a rook's graph is a maximum independent set.
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| ==Characterization==
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| {{harvtxt|Moon|1963}} characterizes the ''m'' × ''n'' rook's graph as the unique graph having the following properties:
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| *It has ''mn'' vertices, each of which is adjacent to ''n'' + ''m'' − 2 edges.
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| *''mn''(''m'' −1)/2 of the edges belong to ''m'' − 2 triangles and the remaining ''mn''(''n'' −1)/2 edges belong to ''n'' − 2 triangles.
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| *Any two vertices that are not adjacent to each other belong to a unique 4-[[cycle (graph theory)|cycle]].
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| When ''n'' = ''m'', these conditions may be abbreviated by stating that an ''n''×''n'' rook's graph is a [[strongly regular graph]] with parameters
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| srg(''n''<sup>2</sup>, 2''n'' − 2, ''n'' − 2, 2), and that every strongly regular graph with these parameters must be an ''n''×''n'' rook's graph when ''n''≠4. When ''n''=4, there is another strongly regular graph, the [[Shrikhande graph]], with the same parameters as the 4×4 rook's graph.
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| The Shrikhande graph can be distinguished from the 4×4 rook's graph in that the [[neighborhood (graph theory)|neighborhood]] of any vertex in the Shrikhande graph is connected in a 6-cycle, while in the rook's graph it is connected into two triangles.
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| == Domination ==
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| The [[domination number]] of a graph is the minimum cardinality among all dominating sets. On the rook's graph a set of vertices is a dominating set if and only if their corresponding squares either occupy, or are a rook's move away from, all squares on the ''m''×''n'' board. For the ''m''×''n'' board the domination number is min(''m'',''n'') {{harv|Yaglom|Yaglom|1987}}.
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| On the rook's graph a ''k''-dominating set is a set of vertices whose corresponding squares attack all other squares (via a rook's move) at least ''k'' times. A ''k''-tuple dominating set on the rook's graph is a set of vertices whose corresponding squares attack all other squares at least ''k'' times and are themselves attacked at least ''k'' − 1 times. The minimum cardinality among all ''k''-dominating and ''k''-tuple dominating sets are the ''k''-domination number and the ''k''-tuple domination number, respectively. On the square board, and for even ''k'', the ''k''-domination number is ''nk''/2 when ''n'' ≥ (''k''<sup>2</sup> − 2''k'')/4 and ''k'' < 2''n''. In a similar fashion, the ''k''-tuple domination number is ''n''(''k'' + 1)/2 when ''k'' is odd and less than ''2n'' {{harv|Burchett|Lane|Lachniet|2009}}.
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| == See also ==
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| * [[King's graph]]
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| * [[Knight's graph]]
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| * [[Lattice graph]]
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| == References ==
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| *{{citation
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| | last = Beka
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| | title = ''K''<sub>''n''</sub>-decomposition of the line graphs of complete bipartite graphs
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| | journal = Octogon Mathematical Magazine
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| | volume = 9
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| | issue = 1A
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| | year = 2001
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| | pages = 135–139
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| | first = Ján}}.
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| *{{cite arxiv
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| | last1 = Bekmetjev | first1 = Airat | last2 = Hurlbert | first2 = Glenn
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| | title = The pebbling threshold of the square of cliques
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| | eprint = math.CO/0406157
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| | year = 2004
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| | class = math.CO}}.
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| *{{cite arxiv
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| | title = Sharp bounds for bandwidth of clique products
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| | eprint = cs.DM/0305051
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| | year = 2003
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| | class = cs.DM}}.
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| *{{citation
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| | last1 = Burchett | first1 = Paul | last2 = Lane | first2 = David | last3 = Lachniet | first3 = Jason
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| | title = K-domination and k-tuple domination on the rook's graph and other results
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| | year = 2009
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| | journal = [[Congressus Numerantium]]
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| | volume = 199
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| | pages = 187–204}}.
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| *{{citation
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| | doi = 10.4007/annals.2006.164.51
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| | last1 = Chudnovsky | first1 = Maria | author1-link = Maria Chudnovsky
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| | authorlink2 = Neil Robertson (mathematician)| last2 = Robertson | first2 = Neil
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| | authorlink3 = Paul Seymour (mathematician) | last3 = Seymour | first3 = Paul
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| | last4 = Thomas | first4 = Robin | author4-link=Robin Thomas (mathematician)
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| | year = 2006
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| | url = http://annals.math.princeton.edu/issues/2006/July2006/ChudnovskyRobertsonSeymourThomas.pdf
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| | title = The strong perfect graph theorem
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| | pages = 51–229
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| }} {{dead link|date=May 2010}}.
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| *{{citation
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| *{{citation
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| | last = Hoffman
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| | title = On the line graph of the complete bipartite graph
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| | journal = [[Annals of Mathematical Statistics]]
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| | pages = 883–885
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| | year = 1964
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| | doi = 10.1214/aoms/1177703593
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| | first = A. J.}}.
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| *{{cite arxiv
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| | title = Random subgraphs of the 2D Hamming graph: the supercritical phase
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| | first1 = Remco | last1 = van der Hofstad | first2 = Malwina J. | last2 = Luczak
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| | year = 2008
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| | eprint = 0801.1607
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| | class = math.PR }}.
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| *{{citation
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| | last1 = Laskar | first1 = Renu | last2 = Wallis | first2 = Charles
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| | title = Chessboard graphs, related designs, and domination parameters
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| *{{cite arxiv
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| | first =Remco |last1=van der Hofstad | first2 = Malwina J. | last2 = Luczak | first3 = Joel | last3 = Spencer | authorlink2 = Joel Spencer
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| | year = 2008
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| | class = math.PR }}.
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| *{{citation
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| | last1 = MacGillivray | first1 = G. | last2 = Seyffarth | first2 = K.
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| | title = Classes of line graphs with small cycle double covers
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| | journal = Australasian Journal of Combinatorics
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| | volume = 24
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| | year = 2001
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| | pages = 91–114}}.
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| *{{citation
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| | last = Moon
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| | journal = [[Annals of Mathematical Statistics]]
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| | volume = 34
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| | issue = 2
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| | year = 1963
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| | first = J. W.}}.
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| *{{citation
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| | last1 = de Werra | first1 = D. | last2 = Hertz | first2 = A.
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| | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]
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| | volume = 195
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| | year = 1999
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| | issue = 1–3
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| | pages = 93–101
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| | url = http://www.gerad.ca/~alainh/Sum.pdf
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| | doi = 10.1016/S0012-365X(98)00168-X}}.
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| *{{citation
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| | last1 = Yaglom | first1 = A. M.
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| | publisher = Dover
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| | title = Challenging Mathematical Problems with Elementary Solutions
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| | year = 1987}}.
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| ==External links==
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| *{{mathworld|title=Lattice Graph|urlname=LatticeGraph}}
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| [[Category:Mathematical chess problems]]
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| [[Category:Perfect graphs]]
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| [[Category:Parametric families of graphs]]
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| [[Category:Regular graphs]]
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