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) = Σ f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by la(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 la d+2. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has la = 4 if not all legs are of odd length. We conjecture that almost all d-leg spiders of size q that satisfies d(d+1) 2(2q-1) with each leg length at least 2 has la = d+1.