Approximate results for rainbow labelings

A simple graph G = ( V , E ) is said to be antimagic if there exists a bijection f : E → [ 1 , | E | ] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f : V → [ 1 , | V | ] ,...

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Vydané v:	Periodica mathematica Hungarica Ročník 74; číslo 1; s. 11 - 21
Hlavní autori:	Lladó, Anna, Miller, Mirka
Médium:	Journal Article Publikácia
Jazyk:	English
Vydavateľské údaje:	Dordrecht Springer Netherlands 01.03.2017 Springer Nature B.V
Predmet:	05 Combinatorics 05C Graph theory 11 Number theory 11C Polynomials and matrices Classificació AMS Combinatòria Decision trees Grafs, Teoria de Graph labeling Graph theory Leaves Matemàtica discreta Matemàtiques i estadística Mathematics Mathematics and Statistics Polinomis Polynomial method Polynomials Teoria de nombres Trees (mathematics) Àlgebra Àrees temàtiques de la UPC Graph labeling Polynomial method
ISSN:	0031-5303, 1588-2829
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Shrnutí:	A simple graph G = ( V , E ) is said to be antimagic if there exists a bijection f : E → [ 1 , \| E \| ] such that the sum of the values of f on edges incident to a vertex takes different values on distinct vertices. The graph G is distance antimagic if there exists a bijection f : V → [ 1 , \| V \| ] , such that ∀ x , y ∈ V , ∑ x i ∈ N ( x ) f x i ≠ ∑ x j ∈ N ( y ) f x j . Using the polynomial method of Alon we prove that there are antimagic injections of any graph G with n vertices and m edges in the interval [ 1 , 2 n + m - 4 ] and, for trees with k inner vertices, in the interval [ 1 , m + k ] . In particular, a tree all of whose inner vertices are adjacent to a leaf is antimagic. This gives a partial positive answer to a conjecture by Hartsfield and Ringel. We also show that there are distance antimagic injections of a graph G with order n and maximum degree Δ in the interval [ 1 , n + t ( n - t ) ] , where t = min { Δ , ⌊ n / 2 ⌋ } , and, for trees with k leaves, in the interval [ 1 , 3 n - 4 k ] . In particular, all trees with n = 2 k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam.
Bibliografia:	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
ISSN:	0031-5303 1588-2829
DOI:	10.1007/s10998-016-0151-2