Local Antimagic Chromatic Number Research Articles

On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles

An edge labeling of a graph G = (V,E) is said to be local antimagic if there 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 is f +(x) = 𝜮 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. For a bipartite circulant graph G, it is known that χ(G)=2 but χla(G) ≥ 3. Moreover, χla(Cn ∨ K1)=3 (respectively 4) if n is even (respectively odd). Let G be a graph of order m ≥ 3. In [Affirmative solutions on local antimagic chromatic number, Graphs Combin., 36 (2020), 1337—1354], the authors proved that if m ≡ n (mod 2) with χla(G) = χ(G), m>n ≥ 4 and m ≥ n2/2, then χla(G∨On) = χla(G)+1. In this paper, we show that the conditions can be omitted in obtaining χla(G∨H) for some circulant graph G, and H is a null graph or a cycle. The local antimagic chromatic number of certain wheel related graphs are also obtained.

On local antimagic chromatic number of cycle-related join graphs II

An edge labeling of a graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices [Formula: see text] and [Formula: see text], [Formula: see text], where the induced vertex label of [Formula: see text] is [Formula: see text] ([Formula: see text] is the set of edges incident to [Formula: see text]). The local antimagic chromatic number of [Formula: see text], denoted by [Formula: see text], is the minimum number of distinct induced vertex labels over all local antimagic labelings of [Formula: see text]. 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.

On the local antimagic chromatic number of the lexicographic product of graphs

On the local antimagic chromatic number of the lexicographic product of graphs

On join product and local antimagic chromatic number of regular graphs

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 $$f \colon E \to \{1,2,\ldots,q\}$$ is called a local antimagic labeling of G if for any two adjacent vertices $$u$$ and $$v$$ , we have $$f^+(u) \ne f^+(v)$$ , where $$f^+(u) = \sum_{e\in E(u)} f(e)$$ , 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 $$\chi_{la}(G)$$ , is the minimum number of induced colors taken over local antimagic labeling of G. Let G and H be two vertex disjoint graphs. The join graph of G and H, denoted $$G \vee H$$ , is the graph with $$V(G\vee H) = V(G) \cup V(H)$$ and $$E(G\vee H) = E(G) \cup E(H) \cup \{uv \mid u\in V(G)$$ , $$v \in V(H)\}$$ . In this paper, we investigated $$\chi_{la}(G\vee H)$$ . Consequently, we show the existence of non-complete regular graphs with arbitrarily large order, regularity and local antimagic chromatic numbers.

Local antimagic chromatic number for the corona product of wheel and null graphs

Let $G=(V,E)$ be a graph of order $p$ and size $q$ having no isolated vertices. A bijection $f\colon E\hm{\rightarrow}\left\{1,2,3,\ldots,q \right\}$ is called a local antimagic labeling if for all $uv\in E$, we have $w(u)\neq w(v)$, the weight $w(u)=\sum_{e\in E(u)}f(e)$, where $E(u)$ is the set of edges incident to $u$. A graph $G$ is local antimagic, if $G$ has a local antimagic labeling. The local antimagic chromatic number $\chi_{la}(G)$ is defined to be the minimum number of colors taken over all colorings of $G$ induced by local antimagic labelings of $G$. In this paper, we completely determine the local antimagic chromatic number for the corona product of wheel and null graphs.

On Local Antimagic Vertex Coloring for Complete Full t-ary Trees

Let G = (V, E) be a finite simple undirected graph without K2 components. A bijection f : E → {1, 2, ⋯, |E|} is called a local antimagic labeling if for any two adjacent vertices u and v, they have different vertex sums, i.e., w(u) ≠ w(v), where the vertex sum w(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 where the vertex v is assigned the color (vertex sum) w(v). The local antimagic chromatic number χla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. It was conjectured [6] that for every tree T the local antimagic chromatic number l + 1 ≤ χla(T) ≤ l + 2, where l is the number of leaves of T. In this article we verify the above conjecture for complete full t-ary trees, for t ≥ 2. A complete full t-ary tree is a rooted tree in which all nodes have exactly t children except leaves and every leaf is of the same depth. In particular we obtain that the exact value for the local antimagic chromatic number of all complete full t-ary trees is l + 1 for odd t.

Approaches that output infinitely many graphs with small local antimagic chromatic number

An edge labeling of a connected graph [Formula: see text] is said to be local antimagic if it is a bijection [Formula: see text] such that for any pair of adjacent vertices x and y, [Formula: see text], where the induced vertex label [Formula: see text], with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by [Formula: see text], is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we show the existence of infinitely many bipartite and tripartite graphs with [Formula: see text].

Local antimagic vertex coloring for generalized friendship graphs

Let G = (V, E) be a graph of order p and size q without isolated vertices. A bijection f: E → {1, 2, … , q} is called a local antimagic labeling if w(u) ≠ w(v) for all uv ∈ E, where the vertex weight w(u) = Ʃ e∈E(u) f(e) and E(u) is the set of edges incident to the vertex u ∈ V. The local antimagic chromatic number χ la (G) is defined to be the minimum number of colors(vertex weights) taken over all colorings of G induced by local antimagic labelings of G. In this paper, we study the local chromatic number for generalized friendship graph of complete graphs and cycles.

On local antimagic chromatic number of spider graphs

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) = Σf(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 first show that a d-leg spider graph has d + 1 ≤ χla ≤ + 2. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has χla = 4 if not all legs are of odd length. No 3-leg spider with all odd leg lengths and χla = 5 is found. This provides partial solutions to the characterization of k-pendant trees T with χla (T) = k + 1 or k + 2. We conjecture that almost all d-leg spiders of size q that satisfy d(d + 1) ≤ 2(2q - 1) with each leg length at least 2 has χla = d + 1.

Local vertex antimagic chromatic number of some wheel related graphs

Let G = (V,E) be a graph of order p and size q having no isolated vertices. A bijection ƒ : E → {1, 2, 3, ..., q} is called a local antimagic labeling if for all uv ∈ E we have w(u) ≠ w(v), the weight w(u) = ∑e∈E(u) f(e) where E(u) is the set of edges incident to u. A graph G is local antimagic if G has a local antimagic labeling. The local antimagic chromatic number χla(G) is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we determine the local antimagic chromatic number for some wheel related graphs.

The Local Antimagic Chromatic Numbers of Some Join Graphs

Let G=(V(G),E(G)) be a connected graph with n vertices and m edges. A bijection f:E(G)→{1,2,⋯,m} is an edge labeling of G. For any vertex x of G, we define ω(x)=∑e∈E(x)f(e) as the vertex label or weight of x, where E(x) is the set of edges incident to x, and f is called a local antimagic labeling of G, if ω(u)≠ω(v) for any two adjacent vertices u,v∈V(G). It is clear that any local antimagic labelling of G induces a proper vertex coloring of G by assigning the vertex label ω(x) to any vertex x of G. The local antimagic chromatic number of G, denoted by χla(G), is the minimum number of different vertex labels taken over all colorings induced by local antimagic labelings of G. In this paper, we present explicit local antimagic chromatic numbers of Fn∨K2¯ and Fn−v, where Fn is the friendship graph with n triangles and v is any vertex of Fn. Moreover, we explicitly construct an infinite class of connected graphs G such that χla(G)=χla(G∨K2¯), where G∨K2¯ is the join graph of G and the complement graph of complete graph K2. This fact leads to a counterexample to a theorem of Arumugam et al. in 2017, and our result also provides a partial solution to Problem 3.19 in Lau et al. in 2021.

On number of pendants in local antimagic chromatic number

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 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. The work of this paper leads to conjecture and problem on graphs with pendant(s).

Local Antimagic Chromatic Number for Copies of Graphs

An edge labeling of a graph G=(V,E) using every label from the set {1,2,⋯,|E(G)|} exactly once is a local antimagic labeling if the vertex-weights are distinct for every pair of neighboring vertices, where a vertex-weight is the sum of labels of all edges incident with that vertex. Any local antimagic labeling induces a proper vertex coloring of G where the color of a vertex is its vertex-weight. This naturally leads to the concept of a local antimagic chromatic number. The local antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local antimagic labelings of G. In this paper, we estimate the bounds of the local antimagic chromatic number for disjoint union of multiple copies of a graph.

On modulo local antimagic chromatic number of graphs

An edge labeling of a connected (p, q) -graph G = (V, E) of order p and size q is a modulo local antimagic labeling if it is a bijection π : E → {1, … ,q} such that for any pair of adjacent vertices u and v, π +(u) ≠ π +(v), where the induced vertex label π +(u) = ∑π(e)(mod p), with e ranging over all the edges incident to u. The modulo local antimagic chromatic number of G, denoted πla (G), is the minimum number of distinct induced vertex labels over all modulo local antimagic labelings of G. In this paper, we shall study the relationship among chromatic number, local antimagic chromatic number and modulo local antimagic chromatic number of graphs. Many open problems for further research are also presented.

On the local antimagic labeling of graphs amalgamation

Let G be a simple connected graph, an ordered pair of sets G(V, E), with V is a set of vertices and E is a set of edges. Graph coloring has been one of the most popular branches in topics of graph theory. In 2017 Arumugam et al. developed a new notion of coloring, namely local antimagic coloring of a graph. This concept is a combination of graph labeling and graph coloring. The local antimagic of graphs is one of the colorings of graph theory that is interesting to study. By the definition, the local antimagic coloring is a bijection f : E(G) ƒ> {1, 2, 3,…, |E(G)|} if for any two adjacent vertices u and v, w(u) = w(v), where w(v) = ΣeeE(v) 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 where the vertex v is assigned the color w(v). The local antimagic chromatic number xla(G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper, we will study the local antimagic coloring of amalgamation of graphs.

The vertex coloring of local antimagic total labeling on corona product graphs

Let G be a graph with the vertex set V(G) and edge set E(G). A function h is a bijective function of domain the union of vertex set and edge set of G and range the natural number {1, 2, 3,… , |V(G)| + |E(G)|. We called the function h as a vertex local antimagic total labeling if for any two adjacent vertices x and x‘, w(x) = w(x’), where w(x) = ∑e∈(x) h(e) + h(x), and E(x) is the set of edges which are incident to x. It is considered to be a proper coloring on vertices of graph G if we assign colour to all vertices with w(x). The minimum number of colors by the vertex local antimagic total labeling of G is called the vertex local antimagic chromatic number, denoted by Xlat(G). We study on the vertex local antimagic total labeling of graphs and determined chromatic number on some corona product graphs, namely corona (G © Cm) where G is isomorphis with path, star, broom, cycle, and sunlet graph.

On the study of local antimagic vertex coloring of graphs and their operations

Arumugam paper studied in et al [4] has inspired us to study the same thing, namely local antimagic edge labeling of graphs. In this paper, we continue to study this type of coloring of some graphs and its operations. By a local antimagic edge labeling, we mean a bijection f from the edge set of G to the natural number up to the number of edges in G such that for any two adjacent vertices v and v’ have different weight, w(v) = w(v’), where w(v) = Σe∈E(v) f (e), and E(v) is the set of vertices which is incident to v. Furthermore, the vertex weight w(v) is assigned as the color on a vertex of G. The local antimagic chromatic number χ1α (G) is the minimum number of colors taken over all coloring induced by vertex local antimagic edge labeling of G.

On local antimagic chromatic number of cycle-related join graphs

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, several sufficient conditions for $\chi_{la}(H)\le \chi_{la}(G)$ 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.

Local antimagic chromatic number of trees - I

Let G = (V, E) be a connected graph with |V|= n and |E|= m. A bijection f : E → {1, 2, … , m} is called a local antimagic lableing if for any two adjacent vertices u and v, w(u) ≠ w(v), where w(u)= ∑ e ϵE(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 where the vertex v is assigned the color w(v). The local antimagic chromatic number χla (G) is the minimum number of colors taken over all colorings induced by local antimagic labelings of G. In this paper we determine the local antimagic chromatic number of several families of trees.

Affirmative Solutions on Local Antimagic Chromatic Number

An edge labeling of a connected graph $$G = (V, E)$$ is said to be local antimagic if it is a bijection $$f:E \rightarrow \{1,\ldots ,|E|\}$$ such that for any pair of adjacent vertices x and y, $$f^+(x)\not = f^+(y)$$ , where the induced vertex label $$f^+(x)= \sum f(e)$$ , with e ranging over all the edges incident to x. The local antimagic chromatic number of G, denoted by $$\chi _{la}(G)$$ , is the minimum number of distinct induced vertex labels over all local antimagic labelings of G. In this paper, we give counterexamples to the lower bound of $$\chi _{la}(G \vee O_2)$$ that was obtained in [Local antimagic vertex coloring of a graph, Graphs Combin. 33:275–285 (2017)]. A sharp lower bound of $$\chi _{la}(G\vee O_n)$$ and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative. We also completely determined the local antimagic chromatic number of complete bipartite graphs.