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Antimagic graph: различия между версиями

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'''Antimagic graph''' --- антимагический граф.
'''Antimagic graph''' — ''[[антимагический граф]].''


An '''antimagic graph''' is a
An '''antimagic graph''' is a [[graph, undirected graph, nonoriented graph|graph]] whose [[edge|edges]] can be labeled with integers <math>1,2, \ldots, e</math> so that the sum of the labels at any given [[vertex]] is different  from  the sum of the labels of any other vertex,  that is,  no two vertices have the same sum. Hartsfield and Ringel conjecture that every [[tree]] other
graph whose edges can be labeled with integers <math>1,2, \ldots, e</math> so
than  <math>\,K_{2}</math>  is antimagic and,  more strongly,  that every [[connected graph]] other than <math>\,K_{2}</math> is antimagic.
that the sum of the labels at any given vertex is different  from  the
sum of the labels of any other vertex,  that is,  no two vertices have
the same sum. Hartsfield and Ringel conjecture that every tree  other
than  <math>K_{2}</math>  is antimagic and,  more strongly,  that every connected
graph other than <math>K_{2}</math> is antimagic.


A special case is an <math>(a,d)</math>-'''antimagic graph'''.  The weight <math>w(x)</math>
A special case is an <math>\,(a,d)</math>-'''antimagic graph'''.  The [[weight (of a vertex)|weight]] <math>\,w(x)</math> of a vertex <math>x \in V(G)</math> under an edge labeling <math>f:  \;  E \rightarrow \{1,2,  \ldots,e\}</math> is the sum of values <math>\,f(xy)</math> assigned to all edges incident with a given vertex <math>\,x</math>.  A connected graph <math>\,G = (V,E)</math> is said to be <math>\,(a,d)</math>-'''antimagic''' if there exist  positive  integers  <math>\,a,d</math> and a bijection <math>f: \; E \rightarrow \{1,2, \ldots, e\}</math> such that the induced mapping <math>g_{f}: \; V(G) \rightarrow W</math> is also bijection,
of a vertex <math>x \in V(G)</math> under an edge labeling <math>f:  \;  E \rightarrow
\{1,2,  \ldots,e\}</math> is the sum of values <math>f(xy)</math> assigned to all edges
incident with a given vertex <math>x</math>.  A connected graph <math>G = (V,E)</math> is said
to be <math>(a,d)</math>-'''antimagic''' if there exist  positive  integers  <math>a,d</math>
and a bijection <math>f: \; E \rightarrow \{1,2, \ldots, e\}</math> such that the
induced mapping <math>g_{f}: \; V(G) \rightarrow W</math> is also bijection,
where <math>W = \{w(x);  x \in V(G)\} = \{a, a+d, a+2d, \ldots, a+(v-1)D\}</math> is
where <math>W = \{w(x);  x \in V(G)\} = \{a, a+d, a+2d, \ldots, a+(v-1)D\}</math> is
the set of the weights of vertices.
the set of the weights of vertices.