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, ff (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, several sufficient conditions for χ la (H) ≤ χ la (G) are obtained, where H is obtained from G with a certain edge deleted or added. We then determined the exact value of the local antimagic chromatic number of many cycle related join graphs.
A total labeling of a graph G = (V, E) is said to be local total antimagic if it is a bijection f : V ∪ E → {1, . . . , |V | + |E|} such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights where the induced weight of a vertex v, w f (v) =f (e) with e ranging over all the edges incident to v, and the induced weight of an edge uv is w f (uv) = f (u) + f (v). The local total antimagic chromatic number of G, denoted by χ lt (G), is the minimum number of distinct induced vertex and edge weights over all local total antimagic labelings of G. In this paper, we first obtained general lower and upper bounds for χ lt (G) and sufficient conditions to construct a graph H with k pendant edges and χ lt (H) ∈ {∆(H) + 1, k + 1}. We then completely characterized graphs G with χ lt (G) = 3. Many families of (disconnected) graphs H with k pendant edges and χ lt (H) ∈ {∆(H) + 1, k + 1} are also obtained.
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.].
Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text], respectively. An edge irregular [Formula: see text]-labeling of [Formula: see text] is a labeling of [Formula: see text] with labels from the set [Formula: see text] in such a way that for any two different edges [Formula: see text] and [Formula: see text], their weights [Formula: see text] and [Formula: see text] are distinct. The weight of an edge [Formula: see text] in [Formula: see text] is the sum of the labels of the end vertices [Formula: see text] and [Formula: see text]. The minimum [Formula: see text] for which the graph [Formula: see text] has an edge irregular [Formula: see text]-labeling is called the edge irregularity strength of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the exact value of edge irregularity strength of corona product of graphs with cycle.
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