Antimagic labelings of caterpillars

Abstract

A k-antimagic labeling of a graph G is an injection from E(G) to {1, 2, . . . , |E(G)| + 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 [13] 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, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use algorithmic aproaches, i.e., constructive approaches, to prove that any caterpillar of order n is ( (n − 1)/2 − 2)-antimagic. Furthermore, if C is a caterpillar with a spine of order s, we prove that when C has at least (3s + 1)/2 leaves or (s − 1)/2 consecutive vertices of degree at most 2 at one end of a longest path, then C is antimagic. As a consequence of a result by Wong and Zhu [22], we also prove that if p is a prime number, any caterpillar with a spine of order p, p − 1 or p − 2 is 1-antimagic. *

List-antimagic labeling of vertex-weighted graphs

“…This was followed by Bérczi et al [BBV15] and Chang et al [CLPZ16] proving that k-regular graphs are antimagic, when k is even and k ≥ 4. Following partial results of Deng and Li [DL19] and Lozano et al [LMS19], Lozano et al [LMST21] proved that all caterpillars are antimagic.…”

Caterpillars are Antimagic

“…In this paper we focus on caterpillars, that is, trees of order at least 3 such that the removal of their leaves produces a path. In [19] the authors give sufficient conditions for a caterpillar to be antimagic and, recently, it has been shown that that caterpillars with maximum degree 3 are antimagic [10]. In this paper we take a step further proving that every caterpillar is antimagic.…”

Caterpillars Have Antimagic Orientations

“…When some specific conditions on the number of leaves or on the vertex degrees are added, caterpillars are known to be antimagic [8,7] but, in the general case, antimagicness is still open for caterpillars. Here we use the flexibility given by the choice of an orientation to adapt the constructive technique from Lozano, Mora, and Seara [8] and prove that all caterpillars admit an antimagic orientation, supporting Conjecture 1.…”