Backbone coloring: различия между версиями
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'''Backbone coloring''' | '''Backbone coloring''' — ''[[хребтовая раскраска]].'' | ||
Consider a graph <math>G = (V,E)</math> with a spanning tree <math>T = (V,E_{T})</math> | Consider a [[graph, undirected graph, nonoriented graph|graph]] <math>\,G = (V,E)</math> with a [[spanning tree]] <math>\,T = (V,E_{T})</math> | ||
(backbone). A vertex coloring <math>f: V \rightarrow \{1,2, \ldots \}</math> is | ([[backbone]]). A [[vertex coloring]] <math>f: V \rightarrow \{1,2, \ldots \}</math> is | ||
proper, if <math>|f(u) - f(v)| \geq 1</math> holds for all edges <math>(u,v) \in E</math>. A | proper, if <math>|f(u) - f(v)| \geq 1</math> holds for all [[edge|edges]] <math>(u,v) \in E</math>. A | ||
vertex coloring is a '''backbone coloring''' for <math>(G,T)</math>, if it is | [[vertex]] coloring is a '''backbone coloring''' for <math>\,(G,T)</math>, if it is | ||
proper and, additionally, <math>|f(u) - f(v)| \geq 2</math> holds for all edges | proper and, additionally, <math>|f(u) - f(v)| \geq 2</math> holds for all edges | ||
<math>(u,v) \in E_{T}</math> in the spanning tree <math>T</math>. | <math>(u,v) \in E_{T}</math> in the spanning [[tree]] <math>\,T</math>. | ||
The '''backbone coloring number''' <math>BBC(G,T)</math> of <math>(G,T)</math> is the | The '''backbone coloring number''' <math>\,BBC(G,T)</math> of <math>\,(G,T)</math> is the | ||
smal-lest integer <math>\ell</math> for which a backbone coloring <math>f: V | smal-lest integer <math>\ell</math> for which a backbone coloring <math>f: V | ||
\rightarrow \{1, 2, \ldots, \ell\}</math> exists. | \rightarrow \{1, 2, \ldots, \ell\}</math> exists. | ||
Версия от 11:53, 12 декабря 2011
Backbone coloring — хребтовая раскраска.
Consider a graph [math]\displaystyle{ \,G = (V,E) }[/math] with a spanning tree [math]\displaystyle{ \,T = (V,E_{T}) }[/math] (backbone). A vertex coloring [math]\displaystyle{ f: V \rightarrow \{1,2, \ldots \} }[/math] is proper, if [math]\displaystyle{ |f(u) - f(v)| \geq 1 }[/math] holds for all edges [math]\displaystyle{ (u,v) \in E }[/math]. A vertex coloring is a backbone coloring for [math]\displaystyle{ \,(G,T) }[/math], if it is proper and, additionally, [math]\displaystyle{ |f(u) - f(v)| \geq 2 }[/math] holds for all edges [math]\displaystyle{ (u,v) \in E_{T} }[/math] in the spanning tree [math]\displaystyle{ \,T }[/math].
The backbone coloring number [math]\displaystyle{ \,BBC(G,T) }[/math] of [math]\displaystyle{ \,(G,T) }[/math] is the smal-lest integer [math]\displaystyle{ \ell }[/math] for which a backbone coloring [math]\displaystyle{ f: V \rightarrow \{1, 2, \ldots, \ell\} }[/math] exists.