Antimagic graph: различия между версиями

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

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

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 (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,