Antimagic labelings of caterpillars

摘要

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 conjectured that every simple connected graph other than K2 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 NullStellenSatz method, to prove our results: (i) any caterpillar of order n is (⌊(n−1)/2⌋−2)-antimagic; (ii) any caterpillar with a spine of order s with either at least ⌊(3s+1)/2⌋ leaves or ⌊(s−1)/2⌋ 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.