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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Bibliographic Details
Published in:Periodica mathematica Hungarica Vol. 74; no. 1; pp. 11 - 21
Main Authors: Lladó, Anna, Miller, Mirka
Format: Journal Article Publication
Language:English
Published: Dordrecht Springer Netherlands 01.03.2017
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:0031-5303
1588-2829
DOI:10.1007/s10998-016-0151-2