On local antimagic chromatic number of graphs with cut-vertices

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. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pendants is obtained. Conjecture 3.1 in [Affirmative solutions on local antimagic chromatic number (2018), submitted] is completely solved. The exact value of the local antimagic chromatic number of many families of graphs with cutvertices are also determined. Consequently, we partially answered Problem 3.1 in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 (2017) 275-285.].

On Local Antimagic Chromatic Number of Spider Graphs

“…Let us analyze the graph G = Sp(2 [n] , 3 [m] ), where (n, m) ∈ A and A is defined in Theorem 2.7. By Appendix, it suffices to consider (n, m) ∈ {(0, 10), (1,8), (1,9), (2, 7), (2, 8), (3,5), (3,6), (4,4), (4,5), (5,3)}.…”

“…The number of distinct induced vertex labels under f is denoted by c(f ), and is called the color number of f . Also, f is call a local antimagic c(f )-labeling of G. The local antimagic chromatic number of G, denoted by χ la (G), is min{c(f ) : f is a local antimagic labeling of G} (see [1][2][3][4][5][6]). For integers a < b, we let [a, b] = {a, a + 1, a + 2, .…”

“…Finally, we show that χ la (Sp(y 1 , y 2 , y 3 )) = 4 if not all y 1 , y 2 , y 3 are odd. We conjecture that each d-leg spider of size q that satisfies d(d + 1) ≤ 2(2q − 1) with each leg length at least 2 has χ la = d + 1 except Sp(2 [n] , 3 [m] ) for (n, m) ∈ {(4, 0), (5, 0), (6, 0), (0, 10), (1,8), (1,9), (2,7), (2,8), (3,5), (3,6), (4,4), (4,5), (5,3)}.…”

Approaches Which Output Infinitely Many Graphs With Small Local Antimagic Chromatic Number

“…We now transform each C 8k and C 8k+4 according that all the published results on 2-connected bipartite graphs have χ la = 2 or 3. Moreover, all the published results on tripartite graphs with at most one pendant have χ la = 3 or 4 (see [1,[3][4][5][6]).…”

On number of pendants in local antimagic chromatic number

“…We conjecture that every tree T k , other than certain caterpillars, with k ≥ 1 pendant vertices has χ la (T k ) = k + 1. The following two results in [7] are needed.…”