On number of pendants in local antimagic chromatic number

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) = ∑f(e), with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by χ_la(G), is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. Let χ(G) be the chromatic number of G. In this paper, sharp upper and lower bounds of χ_la(G) for G with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for k ≥ 1, there are infinitely many graphs with k ≥ χ(G) - 1 pendant vertices and χ_la(G) = k + 1. The work of this paper leads to conjecture and problem on graphs with pendant(s).

Subject Classification:

• 05C78
• 05C69

Keywords:

• Local antimagic labeling
• Local antimagic chromatic number
• Pendant vertices