Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Oriented Coloring of a Grid Abdullah Makkeh Tartu Ülikool October 4, 2015 Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Co-author: Bahman Ghandchi Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References General View Background 1 Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References General View Background 1 Oriented Chromatic number 2 Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References General View Background 1 Oriented Chromatic number 2 Oriented Chromatic number of a grid 3 Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References General View Background 1 Oriented Chromatic number 2 Oriented Chromatic number of a grid 3 Integer programming models 4 Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Notations If G is a graph, we denote by V ( G ) its set of vertices and by E ( G ) its set of edges. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Notations If G is a graph, we denote by V ( G ) its set of vertices and by E ( G ) its set of edges. If G is a digraph, we denote by V ( G ) its set of vertices and by A ( G ) : { ( x , y ) | if the arc is form x to y } its set of arcs. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Notations If G is a graph, we denote by V ( G ) its set of vertices and by E ( G ) its set of edges. If G is a digraph, we denote by V ( G ) its set of vertices and by A ( G ) : { ( x , y ) | if the arc is form x to y } its set of arcs. The order of an undirected graph, or a digraph, is the cardinality of its vertex set. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Notations If G is a graph, we denote by V ( G ) its set of vertices and by E ( G ) its set of edges. If G is a digraph, we denote by V ( G ) its set of vertices and by A ( G ) : { ( x , y ) | if the arc is form x to y } its set of arcs. The order of an undirected graph, or a digraph, is the cardinality of its vertex set. All the graphs we consider are simple and have no loops. Homomorphism Let G and G ′ be two graphs. A homomorphism of G to G ′ is a mapping f : V ( G ) → V ( H ) that preserves the edges: f ( x ) f ( y ) ∈ E ( G ′ ) whenever xy ∈ E ( G ) . Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Notations If G is a graph, we denote by V ( G ) its set of vertices and by E ( G ) its set of edges. If G is a digraph, we denote by V ( G ) its set of vertices and by A ( G ) : { ( x , y ) | if the arc is form x to y } its set of arcs. The order of an undirected graph, or a digraph, is the cardinality of its vertex set. All the graphs we consider are simple and have no loops. Homomorphism Let G and G ′ be two graphs. A homomorphism of G to G ′ is a mapping f : V ( G ) → V ( H ) that preserves the edges: f ( x ) f ( y ) ∈ E ( G ′ ) whenever xy ∈ E ( G ) . If D and D ′ are two digraphs, a homomorphism of D to D ′ is a mapping f : V ( D ) → V ( D ′ ) that preserves the arcs: ( f ( x ) , f ( y )) ∈ E ( D ′ ) whenever ( x , y ) ∈ E ( D ) . Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Graph Coloring A (proper) k -colouring of a graph G is a partition of V ( G ) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Graph Coloring A (proper) k -colouring of a graph G is a partition of V ( G ) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class. χ ( G ) is the chromatic number of G , defined as the smallest k such that G admits a k -colouring. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Graph Coloring A (proper) k -colouring of a graph G is a partition of V ( G ) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class. χ ( G ) is the chromatic number of G , defined as the smallest k such that G admits a k -colouring. Such a k-colouring can be equivalently regarded as a homomorphism of G to the complete graph K k on k vertices. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Graph Coloring A (proper) k -colouring of a graph G is a partition of V ( G ) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class. χ ( G ) is the chromatic number of G , defined as the smallest k such that G admits a k -colouring. Such a k-colouring can be equivalently regarded as a homomorphism of G to the complete graph K k on k vertices. χ ( G ) corresponds to the smallest k such that G admits a homomorphism to K k . Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Graph Coloring A (proper) k -colouring of a graph G is a partition of V ( G ) into k parts, called colour classes, such that no two adjacent vertices belong to the same colour class. χ ( G ) is the chromatic number of G , defined as the smallest k such that G admits a k -colouring. Such a k-colouring can be equivalently regarded as a homomorphism of G to the complete graph K k on k vertices. χ ( G ) corresponds to the smallest k such that G admits a homomorphism to K k . Is there other types of coloring on digraphs? Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Oriented coloring An oriented k-colouring of a digraph D is a partition of V ( D ) into k colour classes such that: Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Oriented coloring An oriented k-colouring of a digraph D is a partition of V ( D ) into k colour classes such that: No two adjacent vertices belong to the same colour class. 1 All the arcs connecting every two colour classes have the same 2 direction. Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Oriented coloring An oriented k-colouring of a digraph D is a partition of V ( D ) into k colour classes such that: No two adjacent vertices belong to the same colour class. 1 All the arcs connecting every two colour classes have the same 2 direction. It is a mapping γ from V ( D ) to a set of k colours such that: Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Oriented coloring An oriented k-colouring of a digraph D is a partition of V ( D ) into k colour classes such that: No two adjacent vertices belong to the same colour class. 1 All the arcs connecting every two colour classes have the same 2 direction. It is a mapping γ from V ( D ) to a set of k colours such that: γ ( u ) �= γ ( v ) for every arc ( u , v ) in A ( D ) . 1 γ ( u ) �= γ ( x ) for every two arcs ( u , v ) and ( w , x ) with γ ( v ) = γ ( w ) . 2 Abdullah Makkeh Oriented Coloring of a Grid
Background Oriented Chromatic number Oriented Chromatic number of a grid Integer programming models References Oriented coloring An oriented k-colouring of a digraph D is a partition of V ( D ) into k colour classes such that: No two adjacent vertices belong to the same colour class. 1 All the arcs connecting every two colour classes have the same 2 direction. It is a mapping γ from V ( D ) to a set of k colours such that: γ ( u ) �= γ ( v ) for every arc ( u , v ) in A ( D ) . 1 γ ( u ) �= γ ( x ) for every two arcs ( u , v ) and ( w , x ) with γ ( v ) = γ ( w ) . 2 The oriented chromatic number χ o ( D ) is the smallest k for which D admits an oriented k -colouring. Abdullah Makkeh Oriented Coloring of a Grid
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