2020/01/15 by Lau, Gee-Choon, Shiu, Wai-Chee, Ng, Ho-Kuen
#05C69 #05C78 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2001.05138
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)\not= 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 χla(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 χla(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 χla(G) = k+1. We conjecture that every tree Tk, other than certain caterpillars, spiders and lobsters, with k≥ 1 pendant vertices has χla(Tk) = k+1.