Approximate results for rainbow labelings - Lladó, Anna; Miller, Mirka (Periodica mathematica Hungarica, 2017)
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.