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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Veröffentlicht in:	Periodica mathematica Hungarica Jg. 74; H. 1; S. 11 - 21
Hauptverfasser:	Lladó, Anna, Miller, Mirka
Format:	Journal Article Verlag
Sprache:	Englisch
Veröffentlicht:	Dordrecht Springer Netherlands 01.03.2017 Springer Nature B.V
Schlagworte:	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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Abstract	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.
AbstractList	The final publication is available at Springer via https://doi.org/10.1007/s10998-016-0151-2] 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, ¿xi¿N(x)f(xi)¿¿xj¿N(y)f(xj). 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 [1,2n+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 ¿ in the interval [1,n+t(n-t)], where t=min{¿,¿n/2¿}, and, for trees with k leaves, in the interval [1,3n-4k]. In particular, all trees with n=2k vertices and no pairs of leaves sharing their neighbour are distance antimagic, a partial solution to a conjecture of Arumugam. Peer Reviewed 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. 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.
Author	Miller, Mirka Lladó, Anna
Author_xml	– sequence: 1 givenname: Anna orcidid: 0000-0002-0993-6556 surname: Lladó fullname: Lladó, Anna email: aina.llado@upc.edu organization: Department of Mathematics, Univ. Politècnica de Catalunya – sequence: 2 givenname: Mirka surname: Miller fullname: Miller, Mirka organization: School of Mathematical and Physical Sciences, University of Newcastle, Department of Mathematics, University of West Bohemia
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Cites_doi	10.1007/BF02773567 10.1017/S0963548398003411 10.1002/jgt.20112 10.1016/j.ejc.2004.04.008 10.1016/j.disc.2007.12.032 10.1016/j.disc.2007.07.027 10.1002/jgt.20027 10.1016/j.tcs.2006.12.003 10.1002/jgt.20347 10.1016/j.tcs.2008.10.023 10.1016/j.disc.2009.09.021 10.1016/j.disc.2008.04.012 10.1007/s11856-012-0020-5 10.1007/11533719_68 10.1007/978-3-642-19222-7_31
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Snippet	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... 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... The final publication is available at Springer via https://doi.org/10.1007/s10998-016-0151-2] A simple graph G=(V,E) is said to be antimagic if there exists a...
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SubjectTerms	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
Title	Approximate results for rainbow labelings
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