Gee-Choon Lau
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View article: On local antimagic chromatic numbers of the join of two special families of graphs
On local antimagic chromatic numbers of the join of two special families of graphs Open
It is known that null graphs are the only (regular) graphs with local antimagic chromatic 1 and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we first use matrices of size \((2m+1) \t…
View article: New families of tripartite graphs with local antimagic chromatic number 3
New families of tripartite graphs with local antimagic chromatic number 3 Open
For a graph \(G=(V,E)\) of size \(q\), a bijection \(f : E \to \{1,2,\ldots,q\}\) is a local antimagic labeling if it induces a vertex labeling \(f^+ : V \to \mathbb{N}\) such that \(f^+(u) \ne f^+(v)\), where \(f^+(u)\) is the sum of all …
View article: Complete characterization of graphs with local total antimagic chromatic number 3
Complete characterization of graphs with local total antimagic chromatic number 3 Open
A total labeling of a graph \(G = (V, E)\) is said to be local total antimagic if it is a bijection \(f: V\cup E \to\{1,\ldots,|V|+|E|\}\) such that adjacent vertices, adjacent edges, and pairs of an incident vertex and edge have distinct …
View article: On Bridge Graphs with Local Antimagic Chromatic Number 3
On Bridge Graphs with Local Antimagic Chromatic Number 3 Open
Let G=(V,E) be a connected graph. A bijection f:E→{1,…,|E|} is called a local antimagic labeling if, for any two adjacent vertices x and y, f+(x)≠f+(y), where f+(x)=∑e∈E(x)f(e), and E(x) is the set of edges incident to x. Thus, a local ant…
View article: On local antimagic chromatic number of the join of two special families of graphs -- II
On local antimagic chromatic number of the join of two special families of graphs -- II Open
It is known that null graphs and 1-regular graphs are the only regular graphs without local antimagic chromatic number. In this paper, we proved that the join of 1-regular graph and a null graph has local antimagic chromatic number is 3. C…
View article: Complete characterization of bridge graphs with local antimagic chromatic number 2
Complete characterization of bridge graphs with local antimagic chromatic number 2 Open
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f\colon E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex …
View article: New Families of tripartite graphs with local antimagic chromatic number 3
New Families of tripartite graphs with local antimagic chromatic number 3 Open
For a graph $G(V,E)$ of size $q$, a bijection $f : E(G) \to [1,q]$ is a local antimagc labeling if it induces a vertex labeling $f^+ : V(G) \to \mathbb{N}$ such that $f^+(u) \ne f^+(v)$, where $f^+(u)$ is the sum of all the incident edge l…
View article: On local antimagic total chromatic number of certain one point union of graphs
On local antimagic total chromatic number of certain one point union of graphs Open
Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A bijection $f:V(G)\cup E(G)\to \{1,2,\ldots,p+q\}$ is called a local antimagic total labeling of $G$ if for any two adjacent vertices $u$ and $v$, we have $w(u)\ne w(v…
View article: On bridge graphs with local antimagic chromatic number 3
On bridge graphs with local antimagic chromatic number 3 Open
Let $G=(V, E)$ be a connected graph. A bijection $f: E\to \{1, \ldots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $x$ and $y$, $f^+(x)\neq f^+(y)$, where $f^+(x)=\sum_{e\in E(x)}f(e)$ and $E(x)$ is the set…
View article: Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach
Construction of local antimagic 3-colorable graphs of fixed even size -- matrix approach Open
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 …
View article: Local distance antimagic cromatic number of join product of graphs with cycles or paths
Local distance antimagic cromatic number of join product of graphs with cycles or paths Open
Let $G$ be a graph of order $p$ without isolated vertices. A bijection $f: V \to \{1,2,3,\dots,p\}$ is called a local distance antimagic labeling, if $w_f(u)\ne w_f(v)$ for every edge $uv$ of $G$, where $w_f(u)=\sum_{x\epsilon N(u)} {f(x)}…
View article: On Local Antimagic Chromatic Number of Graphs with Cut-vertices
On Local Antimagic Chromatic Number of Graphs with Cut-vertices Open
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), wit…
View article: Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach
Constructions of local antimagic 3-colorable graphs of fixed odd size | matrix approach Open
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if there 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 lab…
View article: Complete characterization of graphs with local total antimagic chromatic number 3
Complete characterization of graphs with local total antimagic chromatic number 3 Open
A total labeling of a graph $G = (V, E)$ is said to be local total antimagic if it is a bijection $f: V\cup E \to\{1,\ldots ,|V|+|E|\}$ such that adjacent vertices, adjacent edges, and incident vertex and edge have distinct induced weights…
View article: An algorithmic approach in constructing infinitely many even size graphs with local antimagic chromatic number 3
An algorithmic approach in constructing infinitely many even size graphs with local antimagic chromatic number 3 Open
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 …
View article: Analytic odd mean labeling of union and identification of some graphs
Analytic odd mean labeling of union and identification of some graphs Open
A graph G is analytic odd mean if there exist an injective function f : V → {0, 1, 3, . . . , 2q − 1} with an induced edge labeling f∗ : E → Z such that for each edge uv with f(u) < f(v), is injective. Clearly the values of f∗ are odd. We …
View article: On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles
On local antimagic chromatic numbers of circulant graphs join with null graphs or cycles Open
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) = ? f(e), with …
View article: On bridge graphs with local antimagic chromatic number 3
On bridge graphs with local antimagic chromatic number 3 Open
Let $G=(V, E)$ be a connected graph. A bijection $f: E\to \{1, \ldots, |E|\}$ is called a local antimagic labeling if for any two adjacent vertices $x$ and $y$, $f^+(x)\neq f^+(y)$, where $f^+(x)=\sum_{e\in E(x)}f(e)$ and $E(x)$ is the set…
View article: Sudoku number of graphs
Sudoku number of graphs Open
We introduce a concept in graph coloring motivated by the popular Sudoku puzzle. Let [Formula: see text] be a graph of order n with chromatic number [Formula: see text] and let [Formula: see text] Let [Formula: see text] be a k-coloring of…
View article: On the local antimagic chromatic number of the lexicographic product of graphs
On the local antimagic chromatic number of the lexicographic product of graphs Open
Let G = (V, E) be a connected simple graph.A bijection f : E → {1, 2, . . ., |E|} is said to be a local antimagic labeling of G if f + (u) = f + (v) holds for any two adjacent vertices u and v of G, where E(u) is the set of edges incident …
View article: Every graph is local antimagic total and its applications
Every graph is local antimagic total and its applications Open
Let \(G = (V,E)\) be a simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local antimagic l…
View article: On local antimagic total labeling of complete graphs amalgamation
On local antimagic total labeling of complete graphs amalgamation Open
Let \(G = (V,E)\) be a connected simple graph of order \(p\) and size \(q\). A graph \(G\) is called local antimagic (total) if \(G\) admits a local antimagic (total) labeling. A bijection \(g : E \to \{1,2,\ldots,q\}\) is called a local a…
View article: Complete characterization of s-bridge graphs with local antimagic chromatic number 2
Complete characterization of s-bridge graphs with local antimagic chromatic number 2 Open
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 …
View article: A note on local antimagic chromatic number of lexicographic product graphs
A note on local antimagic chromatic number of lexicographic product graphs Open
Let $G = (V,E)$ be a connected simple graph. A bijection $f: E \rightarrow \{1,2,\ldots,|E|\}$ is called a local antimagic labeling of $G$ if $f^+(u) \neq f^+(v)$ holds for any two adjacent vertices $u$ and $v$, where $f^+(u) = \sum_{e\in …
View article: Sudoku Number of Graphs
Sudoku Number of Graphs Open
We introduce a new concept in graph coloring motivated by the popular Sudoku puzzle. Let $G=(V,E)$ be a graph of order $n$ with chromatic number $χ(G)=k$ and let $S\subseteq V.$ Let $\mathscr C_0$ be a $k$-coloring of the induced subgraph …
View article: On local antimagic chromatic number of lexicographic product graphs
On local antimagic chromatic number of lexicographic product graphs Open
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 $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if …
View article: On local antimagic chromatic number of a corona product graph
On local antimagic chromatic number of a corona product graph Open
In this paper, we provide a correct proof for the lower bounds of the local antimagic chromatic number of the corona product of friendship and fan graphs with null graph respectively as in [On local antimagic vertex coloring of corona prod…
View article: On join product and local antimagic chromatic number of regular graphs
On join product and local antimagic chromatic number of regular graphs Open
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 $f : E \to \{1,2,\ldots,q\}$ is called a local antimagic labeling of $G$ if …
View article: On local antimagic total labeling of amalgamation graphs
On local antimagic total labeling of amalgamation graphs Open
Let $G = (V,E)$ be a connected simple graph of order $p$ and size $q$. A graph $G$ is called local antimagic (total) if $G$ admits a local antimagic (total) labeling. A bijection $g : E \to \{1,2,\ldots,q\}$ is called a local antimagic lab…
View article: Reflexive edge strength of convex polytopes and corona product of cycle with path
Reflexive edge strength of convex polytopes and corona product of cycle with path Open
For a graph $ G $, we define a total $ k $-labeling $ \varphi $ is a combination of an edge labeling $ \varphi_e(x)\to\{1, 2, \ldots, k_e\} $ and a vertex labeling $ \varphi_v(x) \to \{0, 2, \ldots, 2k_v\} $, such that $ \varphi(x) = \varp…