Antimagic labelings of caterpillars

A k-antimagic labeling of a graph G is an injection from E(G) to {1, 2, ..., vertical bar E(G)vertical bar +k} such that all vertex sums are pairwise distinct, where the vertex sum at vertex u is the sum of the labels assigned to edges incident to u. We call a graph k-antimagic when it has a k-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than K-2 is antimagic, but the conjecture is still open even for trees. Here we study k-antimagic labelings of caterpillars. We use algorithmic and constructive techniques, instead of the standard Combinatorial NullStel-lenSatz method, to prove our results: (i) any caterpillar of order n is ( left perpendicular n- 1)/2 right perpendicular - 2)- antimagic; (ii) any caterpillar with a spine of order s with either at least left perpendicular (3s + 1)/2 right perpendicular leaves or left perpendicular(s - 1)/2 right perpendicular consecutive vertices of degree at most 2 at one end of a longest path, is antimagic; and (iii) if p is a prime number, any caterpillar with a spine of order p, p -1 or p - 2 is 1-antimagic. (C) 2018 Elsevier Inc. All rights reserved.