2020/09/04 by Lau, Gee-Choon, Li, Jianxi, Ng, Ho-Kuen +1
#05C69 #05C78 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2009.01996
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. In this paper, we (i) give a sufficient condition for a graph with one pendant to have χla≥ 3. A necessary and sufficient condition for a graph to have χla=2 is then obtained; (ii) give a sufficient condition for every circulant graph of even order to have χla = 3; (iii) construct infinitely many bipartite and tripartite graphs with χla = 3 by transformation of cycles; (iv) apply transformation of cycles to obtain infinitely many one-point union of regular (possibly circulant) or bi-regular graphs with χla = 2,3. The work of this paper suggests many open problems on the local antimagic chromatic number of bipartite and tripartite graphs.