On local antimagic chromatic number of spider graphs

Abstract

An edge labeling of a connected graph G = (V, E) is said to be local antimagic if it is a Bijection f : E → {1, … ,|E|} such that for any pair of adjacent vertices x and y, f +(x) ≠ f +(y), where the induced vertex label f +(x) = Σf(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χla (G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we first show that a d-leg spider graph has d + 1 ≤ χla ≤ + 2. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has χla = 4 if not all legs are of odd length. No 3-leg spider with all odd leg lengths and χla = 5 is found. This provides partial solutions to the characterization of k-pendant trees T with χla (T) = k + 1 or k + 2. We conjecture that almost all d-leg spiders of size q that satisfy d(d + 1) ≤ 2(2q - 1) with each leg length at least 2 has χla = d + 1.