Local Antimagic Chromatic Number for Copies of Graphs

arXiv (Cornell University), 2023

Consider a simple graph G without K 2 component with vertex set V and edge set E. Local antimagic labeling f of G is a one-to-one mapping of edges to distinct positive integers 1, 2, . . . , |E| such that the weights of adjacent vertices are distinct, where the weight of a vertex is the sum of labels assigned to the edges incident to it. These weights of the vertex induced by local antimagic labeling result in a proper vertex coloring of the graph G. We define the local antimagic chromatic number of G, denoted as χ la (G), as the smallest number of distinct weights obtained across all possible local antimagic labelings of G. In this paper, we explore the local antimagic chromatic numbers of various classes of graphs, including the union of certain graph families, the corona product of graphs, and the necklace graph. Additionally, we provide constructions for infinitely many graphs for which χ la (G) equals the chromatic number χ(G) of the graph.

Acta Mathematica Hungarica, 2023

Let G = (V, E) be a connected simple graph of order p and size q. A graph G is called local antimagic if G admits a local antimagic labeling. A bijection f : E → {1, 2,. .. , q} is called a local antimagic labeling of G if for any two adjacent vertices u and v, we have f + (u) = f + (v), where f + (u) = e∈E(u) f (e), and E(u) is the set of edges incident to u. Thus, any local antimagic labeling induces a proper vertex coloring of G if vertex v is assigned the color f + (v). The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The lexicographic product of G and H, denoted G[H], is the graph with vertex set V (G) × V (H), and (u, u ′) is adjacent to (v, v ′) in G[H] if (u, v) ∈ E(G) or if u = v and u ′ v ′ ∈ E(H). In this paper, we obtained sharp upper bound of χ la (G[O n ]) where O n is a null graph of order n ≥ 1. Sufficient conditions for even regular bipartite and tripartite graphs G to have χ la (G) = 3 are also obtained. Consequently, we successfully determined the local antimagic chromatic number of infinitely many (connected and disconnected) regular graphs that partially support the existence of r-regular graph G of order p such that (i) χ la (G) = χ(G) = k, and (ii) χ la (G) = χ(G) + 1 = k for each possible r, p, k.

AIP Conference Proceedings, 2018

A total labeling of simple and connected graph G(V, E) is said to be local antimagic total edge labeling if a bijection f : V(G) ∪ E(G) −→ {1, 2, 3, ..., |V(G)| + |E(G)|}, w t (e 1) w t (e 2) for any two adjacent edges e 1 and e 2 , where for e = uv ∈ G, w t (e) = f (u) + f (uv) + f (v). The local antimagic total edge labeling induces a proper edge coloring of G if each edge e is assigned the color w t (e). The minimum number of local antimagic total edge chromatic number of G denoted by γ leat (G), is the distinct induced by edge labels over all local antimagic total labeling of G. In this paper we study the existence of local edge antimagic total chromatic number of amalgamation of some special graphs namely amal(P n , v, m), amal(C n , v, m),amal(F n , v, m) and amal(S n , v, m).

Electronic Journal of Graph Theory and Applications

Let G(V, E) be a simple graph and f be a bijection f : V ∪ E → {1, 2,. .. , |V | + |E|} where f (V) = {1, 2,. .. , |V |}. For a vertex x ∈ V , define its weight w(x) as the sum of labels of all edges incident with x and the vertex label itself. Then f is called a super vertex local antimagic total (SLAT) labeling if for every two adjacent vertices their weights are different. The super vertex local antimagic total chromatic number χ slat (G) is the minimum number of colors taken over all colorings induced by super vertex local antimagic total labelings of G. We classify all trees T that have χ slat (T) = 2, present a class of trees that have χ slat (T) = 3, and show that for any positive integer n ≥ 2 there is a tree T with χ slat (T) = n.

Symmetry

Let G=(V,E) be a graph with vertex set V and edge set E. A local antimagic total vertex coloring f of a graph G with vertex-set V and edge-set E is an injective map from V∪E to {1,2,…,|V|+|E|} such that if for each uv∈E(G) then w(u)≠w(v), where w(u)=∑uv∈E(G)f(uv)+f(u). If the range set f satisfies f(V)={1,2,…,|V|}, then the labeling is said to be local super antimagic total labeling. This labeling generates a proper vertex coloring of the graph G with the color w(v) assigning the vertex v. The local super antimagic total chromatic number of graph G, χlsat(G) is defined as the least number of colors that are used for all colorings generated by the local super antimagic total labeling of G. In this paper we investigate the existence of the local super antimagic total chromatic number for some particular classes of graphs such as a tree, path, cycle, helm, wheel, gear, sun, and regular graphs as well as an amalgamation of stars and an amalgamation of wheels.

arXiv (Cornell University), 2022

Let G = (V, E) be a connected simple graph. A bijection f : E → {1, 2,. .. , |E|} is called a local antimagic labeling of G if f + (u) = f + (v) holds for any two adjacent vertices u and v, where f + (u) = e∈E(u) f (e) and E(u) is the set of edges incident to u. A graph G is called local antimagic if G admits at least a local antimagic labeling. The local antimagic chromatic number, denoted χ la (G), is the minimum number of induced colors taken over local antimagic labelings of G. Let G and H be two disjoint graphs. The graph G[H] is obtained by the lexicographic product of G and H. In this paper, we obtain sufficient conditions for χ la (G[H]) ≤ χ la (G)χ la (H). Consequently, we give examples of G and H such that χ la (G[H]) = χ(G)χ(H), where χ(G) is the chromatic number of G. We conjecture that (i) there are infinitely many graphs G and H such that χ la (G[H]) = χ la (G)χ la (H) = χ(G)χ(H), and (ii) for k ≥ 1, χ la (G[H]) = χ(G)χ(H) if and only if χ(G)χ(H) = 2χ(H) + ⌈ χ(H) k ⌉, where 2k + 1 is the length of a shortest odd cycle in G.

2018

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 cut-vertices are also determined. Consequently, several families of G such that χ_la(G) = |V(G)| are also obtained.

IOP Conference Series: Earth and Environmental Science, 2019

A graph G = (V, E) be connected, finite, and undirected graph without multiple edges and loops. Graph G(V, E) consists two sets of vertices V and edge E. A graph called related cycle if the subgraph of graph G contains a cycle. Local antimagic total edge labeling is defined a bijection f : V ∪ E −→ {1, 2, 3, ..., |V | + |E|}, if for any two adjacent edges e1 and e2, wt(e1) = wt(e2), where for e = ab ∈ G, wt(e) = f (a) + f (ab) + f (b). Thus, the local antimagic total edge labeling induces a proper edge coloring of G if each edge e is assigned the color wt(e). The local antimagic total edge chromatic number of G denoted by γ late (G), is the minimum of colors needed to coloring the edges of a graph, the number of distinct induced edge labels over all local antimagic total labeling of G. In this paper we study the local antimagic total edge coloring of some related cycle of graphs and determined the chromatic number of some related cycle graphs namely triangular book Btr, prism graph Pr, sun graph Mr and kite graph Kr,s.

Journal of Physics: Conference Series, 2018

All graph in this paper are finite, simple and connected graph. Let G(V, E) be a graph of vertex set V and edge set E. A bijection f : V (G) −→ {1, 2, 3, ..., |V (G)|} is called a local edge antimagic labeling if for any two adjacent edges e 1 and e 2 , w(e 1) = w(e 2), where for e = uv ∈ G, w(e) = f (u) + f (v). Thus, any local edge antimagic labeling induces a proper edge coloring of G if each edge e is assigned the color w(e). The local edge antimagic hromatic number γ lea (G) is the minimum number of colors taken over all colorings induced by local edge antimagic labelings of G. In this paper, we have found the lower bound of the local edge antimagic coloring of G H and determine exact value local edge antimagic coloring of G H.

2021

An edge labeling of a 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 of x is f^+(x)= ∑_e∈ E(x) f(e) (E(x) is the set of 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 to determine the local antimagic chromatic number of the join of graphs are obtained. We then determine the exact value of the local antimagic chromatic number of many join graphs.