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Gee-Choon Lau

Publications and source records attributed to Gee-Choon Lau.

31 records · Page 2Linked to original sources

arXiv2022

On local antimagic chromatic number of graphs with cut-vertices

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 $χ_{la}(G)$, 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. The exact value of the local antimagic chromatic number of many families of graphs with cut-vertices (possibly given by pendant edges) are also determined. Consequently, we partially answered Problem 3.1 in [Local antimagic vertex coloring of a graph, {\it Graphs and Combin.}, {\bf33} (2017), 275--285.].

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arXiv2021

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

An edge labeling of a 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 of $x$ is $f^+(x)= \sum_{e\in E(x)} f(e)$ ($E(x)$ 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.

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arXiv2021

Further Results On k-Super Graceful Graphs

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful labeling. In this paper, we study the $k$-super gracefulness of some graphs in which each component is either regular or bi-regular.

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arXiv2021

Graphs With Minimal Strength

For any graph $G$ of order $p$, a bijection $f: V(G)\to [1,p]$ is called a numbering of the graph $G$ of order $p$. The strength $str_f(G)$ of a numbering $f: V(G)\to [1,p]$ of $G$ is defined by $str_f(G) = \max\{f(u)+f(v)\; |\; uv\in E(G)\},$ and the strength $str(G)$ of a graph $G$ itself is $str(G) = \min\{str_f(G)\;|\; f \mbox{ is a numbering of } G\}.$ A numbering $f$ is called a strength labeling of $G$ if $str_f(G)=str(G)$. In this paper, we obtained a sufficient condition for a graph to have $str(G)=|V(G)|+\d(G)$. Consequently, many questions raised in [Bounds for the strength of graphs, {\it Aust. J. Combin.} {\bf72(3)}, (2018) 492--508] and [On the strength of some trees, {\it AKCE Int. J. Graphs Comb.} (Online 2019) doi.org/10.1016/j.akcej.2019.06.002] are solved. Moreover, we showed that every graph $G$ either has $str(G)=|V(G)|+\d(G)$ or is a proper subgraph of a graph $H$ that has $str(H) = |V(H)| + \d(H)$ with $\d(H)=\d(G)$. Further, new good lower bounds of $str(G)$ are also obtained. Using these, we determined the strength of 2-regular graphs and obtained new lower bounds of $str(Q_n)$ for various $n$, where $Q_n$ is the $n$-regular hypercube.

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arXiv2020

Every Graph Is Local Antimagic Total And Its Applications To Local Antimagic (Total) Chromatic Numbers

A graph $G = (V, E)$ of order $p$ and size $q$ is said to be local antimagic if there exists a bijection $g:E(G) \to \{1,2,\ldots,q\}$ such that for any pair of adjacent vertices $u$ and $v$, $g^+(u)\ne g^+(v)$, where $g^+(u)=\sum_{uv\in E(G)} g(uv)$ is the induced vertex color of $u$ under $g$. We also say $G$ is local antimagic total if there exists a bijection $f: V\cup E \to\{1,2,\ldots ,p+q\}$ such that for any pair of adjacent vertices $u$ and $v$, $w(u)\not= w(v)$, where $w(u)= f(u) +\sum_{uv\in E(G)} f(uv)$ is the induced vertex weight of $u$ under $f$. The local antimagic (and local antimagic total) chromatic number of $G$, denoted $χ_{la}(G)$ (and $χ_{lat}(G)$), is the minimum number of distinct induced vertex colors (and weights) over all local antimagic (and local antimagic total) labelings of $G$. We also say a local antimagic total labeling is local super antimagic total if $f(v)\in\{1,2,\ldots,p\}$ for each $v\in V(G)$. In [Proof of a local antimagic conjecture, {\it Discrete Math. Theor. Comp. Sc.}, {\bf 20(1)} (2018), \#18], the author proved that every connected graph of order at least 3 is local antimagic. Using this result, we provide a very short proof that every graph is local antimagic total. We showed that there exists close relationship between $χ_{la}(G \vee K_1)$ and $χ_{lat}(G)$. A sufficient condition is also given for the corresponding local super antimagic total labeling. Sharp bounds of $χ_{lat}(G)$ and close relationships between $χ_{lat}(G)$ and $χ_{la}(G \vee K_1)$ are found. Bounds of $χ_{lat}(G-e)$ in terms of $χ_{lat}(G)$ for a graph $G$ with an edge $e$ deleted are also obtained. These relationships are used to determine the exact values of $χ_{lat}$ for many graphs $G$. We also conjecture that each graph $G$ of order at least 3 has $χ_{lat}(G)\le χ_{la}(G)$.

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arXiv2020

Approaches Which Output Infinitely Many Graphs With Small 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 $f:E \to\{1,\ldots ,|E|\}$ 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 $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we (i) give a sufficient condition for a graph with one pendant to have $χ_{la}\ge 3$. A necessary and sufficient condition for a graph to have $χ_{la}=2$ is then obtained; (ii) give a sufficient condition for every circulant graph of even order to have $χ_{la} = 3$; (iii) construct infinitely many bipartite and tripartite graphs with $χ_{la} = 3$ by transformation of cycles; (iv) apply transformation of cycles to obtain infinitely many one-point union of regular (possibly circulant) or bi-regular graphs with $χ_{la} = 2,3$. The work of this paper suggests many open problems on the local antimagic chromatic number of bipartite and tripartite graphs.

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arXiv2020

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 $f : E \to \{1, . . . , |E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x) \ne f^+(y)$, 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 $χ_{la}(G)$, 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 $d+1\le χ_{la}\le d+2$. We then obtain many sufficient conditions such that both the values are attainable. Finally, we show that each 3-leg spider has $χ_{la} = 4$ if not all legs are of odd length. We conjecture that almost all $d$-leg spiders of size $q$ that satisfies $d(d+1) \le 2(2q-1)$ with each leg length at least 2 has $χ_{la} = d+1$.

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arXiv2020

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 $f:E \to\{1,\ldots ,|E|\}$ 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 $χ_{la}(G)$, 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 $χ_{la}(G \vee O_2)$ that was obtained in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 : 275 - 285 (2017)]. A sharp lower bound of $χ_{la}(G\vee O_n)$ 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.

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arXiv2020

On Integer Sequences Associated To Two Distinct Sums

In this note, we show the existence of integer sequences of lengths at least 3 (except 7) such that for every integer in position $i\equiv 1\pmod{4}$ (respectively position $j\equiv 3\pmod{4}$), counting from left to right, the sum of the integer and the adjacent integer(s) has a constant sum $x$ (respectively $y$) with $x\ne y$.

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arXiv2020

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 $f:E \to\{1,\ldots ,|E|\}$ 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 $χ_{la}(G)$, 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 $χ_{la}(G)$ for $G$ with pendant vertices, and sufficient conditions for the bounds to equal, are obtained. Consequently, for $k\ge 1$, there are infinitely many graphs with $k \ge χ(G) - 1$ pendant vertices and $χ_{la}(G) = k+1$. We conjecture that every tree $T_k$, other than certain caterpillars, spiders and lobsters, with $k\ge 1$ pendant vertices has $χ_{la}(T_k) = k+1$.

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arXiv2018

Cartesian Magicness of 3-Dimensional Boards

A $(p,q,r)$-board that has $pq+pr+qr$ squares consists of a $(p,q)$-, a $(p,r)$-, and a $(q,r)$-rectangle. Let $S$ be the set of the squares. Consider a bijection $f : S \to [1,pq+pr+qr]$. Firstly, for $1 \le i \le p$, let $x_i$ be the sum of all the $q+r$ integers in the $i$-th row of the $(p,q+r)$-rectangle. Secondly, for $1 \le j \le q$, let $y_j$ be the sum of all the $p+r$ integers in the $j$-th row of the $(q,p+r)$-rectangle. Finally, for $1\le k\le r$, let $z_k$ be the the sum of all the $p+q$ integers in the $k$-th row of the $(r,p+q)$-rectangle. Such an assignment is called a $(p,q,r)$-design if $\{x_i : 1\le i\le p\}=\{c_1\}$ for some constant $c_1$, $\{y_j : 1\le j\le q\}=\{c_2\}$ for some constant $c_2$, and $\{z_k : 1\le k\le r\}=\{c_3\}$ for some constant $c_3$. A $(p,q,r)$-board that admits a $(p,q,r)$-design is called (1) Cartesian tri-magic if $c_1$, $c_2$ and $c_3$ are all distinct; (2) Cartesian bi-magic if $c_1$, $c_2$ and $c_3$ assume exactly 2 distinct values; (3) Cartesian magic if $c_1 = c_2 = c_3$ (which is equivalent to supermagic labeling of $K(p,q,r)$). Thus, Cartesian magicness is a generalization of magic rectangles into 3-dimensional space. In this paper, we study the Cartesian magicness of various $(p,q,r)$-board by matrix approach involving magic squares or rectangles. In Section~2, we obtained various sufficient conditions for $(p,q,r)$-boards to admit a Cartesian tri-magic design. In Sections~3 and~4, we obtained many necessary and (or) sufficient conditions for various $(p,q,r)$-boards to admit (or not admit) a Cartesian bi-magic and magic design. In particular, it is known that $K(p,p,p)$ is supermagic and thus every $(p,p,p)$-board is Cartesian magic. We gave a short and simpler proof that every $(p,p,p)$-board is Cartesian magic.

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arXiv2018

On local antimagic chromatic number of cycle-related join graphs

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arXiv2012

On the k-edge magic graphs

Let $G$ be a graph with vertex set V and edge set E such that |V| = p and |E| = q. For integers k\geq 0, define an edge labeling f : E \rightarrow \{k,k+1,....,k+q-1\} and define the vertex sum for a vertex $v$ as the sum of the labels of the edges incident to v. If such an edge labeling induces a vertex labeling in which every vertex has a constant vertex sum (mod p), then G is said to be k-edge magic (k-EM). In this paper, we (i) show that all the maximal outerplanar graphs of order p = 4; 5; 7 are k-EM if and only if k\equiv 2 (mod p); (ii) obtain all the maximal outerplanar graphs that are k-EM for k = 3; 4; and (iii) characterize all (p; p-h)-graph that are k-EM for h\geq 0. We conjecture that all maximal outerplanar graphs of prime order p are k-EM if and only if k \equiv 2 (mod p).

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