The Integer-antimagic Spectra of Graphs with a Chord

Abstract

Let A be a nontrival abelian group. A connected simple graph G = (V, E) is Aantimagic if there exists an edge labeling f : E(G) → A \ {0} such that the induced vertex labeling f + : V (G) → A, defined by f + (v) = uv∈E(G) f (uv), is injective. The integer-antimagic spectrum of a graph G is the set IAM(G) = {k | G is Z k-antimagic and k ≥ 2}. In this paper, we determine the integer-antimagic spectra for cycles with a chord, paths with a chord, and wheels with a chord.

The integer-antimagic spectra of Hamiltonian graphs

“…Conjecture 1 was shown to be true for all of the classes of graphs which were analyzed in [1,4,5,7,8,9]. The purpose of this paper is to provide additional evidence for Conjecture 1 by verifying it for a large family of graphs including all Hamiltonian graphs.…”

“…We call the shorter of the two cycles the minor subcycle of C m (l), denoted by C − m (l), and the longer of the two cycles the major subcycle of C m (l), denoted by C + m (l). In [4], the integer-antimagic spectrum for cycles with a chord was determined completely. In Lemma 2.1, it was shown that Z k -antimagicness can be preserved when an edge is added, provided that edge lies on an even cycle.…”

“…There is a large body of research on A-magic graphs within the mathematical literature. As for A-antimagic graphs (which is the focus of our paper), cycles, paths, various classes of trees, dumbbells, graphs with a chord, multi-cyclic graphs, K m,n , K m,n − {e}, tadpoles and lollipop graphs were investigated in [1,3,4,5,7,8,9].…”