• Journal of the Korean Institute of Intelligent Systems
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• v.21 no.1
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• pp.138-144
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• 2011

L(2, 1)-labeling problem of a graph G = (V, E) is a problem to find an efficient way to distribute radio frequencies to various wireless equipments in wireless networks. In this work, we suggest a Simulated Annealing algorithm that can be applied to the L(2, 1)-labeling problem. By applying the suggested algorithm to various graphs we will try to show the efficiency of our algorithm.

• The KIPS Transactions:PartB
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• v.15B no.2
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• pp.131-136
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• 2008

L(2,1)-labeling of a graph G is a function f: V(G) $\rightarrow$ {0, 1, 2, ...} such that $|f(u)\;-\;f(\upsilon)|\;{\geq}\;2$ when d(u, v) = 1 and $|f(u)\;-\;f(\upsilon)|\;{\geq}\;1$ when d(u, $\upsilon$) = 2. L(2,1)-labeling number of G, denoted by ${\lambda}(G)$, is the smallest number m such that G has an L(2,1)-labeling with no label greater than m. Since this problem has been proved to be NP-complete, in this article, we develop genetic algorithms for L(2,1)-labeling problem and show that the suggested genetic algorithm peforms very efficiently by applying the algorithms to the class of graphs with known optimum values.

• Journal of applied mathematics & informatics
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• v.41 no.3
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• pp.511-524
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• 2023

An L(p₁, p₂, p₃, . . . , p_m)-labeling of a graph G is an assignment of non-negative integers, called as labels, to the vertices such that the vertices at distance i should have at least pi as their label difference. If p₁ = 4, p₂ = 3, p₃ = 2, p₄ = 1, then it is called a L(4, 3, 2, 1)-labeling which is widely studied in the literature. A L(4, 3, 2, 1)-path coloring of graphs, is a labeling g : V (G) → Z⁺ such that there exists at least one path P between every pair of vertices in which the labeling restricted to this path is a L(4, 3, 2, 1)-labeling. This concept was defined and results for some simple graphs were obtained by the same authors in an earlier article. In this article, we study the concept of L(4, 3, 2, 1)-path coloring for complete bipartite graphs, 2-edge connected split graph, Cartesian product and join of two graphs and prove an existence theorem for the same.

• Korean Journal of Mathematics
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• v.25 no.2
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• pp.279-301
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• 2017

An L(3, 2, 1)-labeling for the graph G = (V, E) is an assignment f of a label to each vertices of G such that ${\mid}f(u)-f({\upsilon}){\mid}{\geq}4-k$ when $dist(u,{\upsilon})=k{\leq}3$. The L(3, 2, 1)-labeling number, denoted by ${\lambda}_{3,2,1}(G)$, for G is the smallest number N such that there is an L(3, 2, 1)-labeling for G with span N. In this paper, we compute the L(3, 2, 1)-labeling number ${\lambda}_{3,2,1}(G)$ when G is a cylindrical grid, which is the cartesian product $P_m{\Box}C_n$ of the path and the cycle, when $m{\geq}4$ and $n{\geq}138$. Especially when n is a multiple of 4, or m = 4 and n is a multiple of 6, then we have ${\lambda}_{3,2,1}(G)=11$. Otherwise ${\lambda}_{3,2,1}(G)=12$.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.119-139
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• 2022

This paper deals with the λ-labeling and L(2, 1)-coloring of simple graphs. A λ-labeling of a graph G is any labeling of the vertices of G with different labels such that any two adjacent vertices receive labels which differ at least two. Also an L(2, 1)-coloring of G is any labeling of the vertices of G such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial λ-labeling f is given in a graph G. A general question is whether f can be extended to a λ-labeling of G. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of G. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in L(2, 1)-coloring and λ-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph K_n☐K_n and the generation of λ-squares.

• Journal of Applied and Pure Mathematics
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• v.5 no.1_2
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• pp.41-53
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• 2023

Let G = (V, E) be a (p, q) graph. Define $${\rho}=\{{\frac{2}{p}},\;{\text{{\qquad} if p is even}}\\{\frac{2}{p-1}},\;{{\text{if p is odd}}$$ and L = {±1, ±2, ±3, … , ±ρ} called the set of labels. Consider a mapping f : V ⟶ L by assigning different labels in L to the different elements of V when p is even and different labels in L to p-1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that ${\mid}{\Delta}_{f_1}-{\Delta}_{f^c_1}{\mid}{\leq}1$, where ${\Delta}_{f_1}$ and ${\Delta}_{f^c_1}$ respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate pair difference cordial labeling behaviour of Petersen graphs P(n, k) like P(n, 2), P(n, 3), P(n, 4).

• Journal of applied mathematics & informatics
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• v.42 no.1
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• pp.1-14
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• 2024

Let G = (V, E) be a (p, q) graph. Define $$\begin{cases}\frac{p}{2},\:if\:p\:is\:even\\\frac{p-1}{2},\:if\:p\:is\:odd\end{cases}$$ and L = {±1, ±2, ±3, ···, ±ρ} called the set of labels. Consider a mapping f : V → L by assigning different labels in L to the different elements of V when p is even and different labels in L to p - 1 elements of V and repeating a label for the remaining one vertex when p is odd.The labeling as defined above is said to be a pair difference cordial labeling if for each edge uv of G there exists a labeling |f(u) - f(v)| such that |Δ_f1 - Δ_f^c1| ≤ 1, where Δ_f1 and Δ_f^c1 respectively denote the number of edges labeled with 1 and number of edges not labeled with 1. A graph G for which there exists a pair difference cordial labeling is called a pair difference cordial graph. In this paper we investigate the pair difference cordial labeling behavior of subdivision of some graphs.

• Korean Journal of Mathematics
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• v.23 no.4
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• pp.607-618
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• 2015

An L(2; 1)-labeling of a graph G is a function f from the vertex set V (G) to the set of all non-negative integers such that ${\mid}f(u)-f(v){\mid}{\geq}2$ if d(u, v) = 1 and ${\mid}f(u)-f(v){\mid}{\geq}1$ if d(u, v) = 2. The ${\lambda}$-number of G, denoted ${\lambda}(G)$, is the smallest number k such that G admits an L(2, 1)-labeling with $k=\max\{f(u){\mid}u{\in}V(G)\}$. In this paper, we consider the square of a cycle and provide exact value for its ${\lambda}$-number. In addition, we also completely determine its edge span.

• The Transactions of the Korea Information Processing Society
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• v.6 no.8
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• pp.2124-2132
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• 1999

Given a graph G=(V,E), Ld(2,1)-labeling of G is a function f : V(G)$\longrightarrow$[0,$\infty$) such that, if v1,v2$\in$V are adjacent, $\mid$ f(x)-f(y) $\mid$$\geq$2d, and, if the distance between and is two, $\mid$ f(x)-f(y) $\mid$$\geq$d, where dG(,v2) is shortest distance between v1 and in G. The L(2,1)-labeling number (G) is the smallest number m such that G has an L(2,1)-labeling f with maximum m of f(v) for v$\in$V. This problem has been studied by Griggs, Yeh and Sakai for the various classes of graphs. In this paper, we discuss the upper-bound of ${\lambda}$ (G) for a chordal graph G and that of ${\lambda}$(G') for a permutation graph G'.

• Korean Journal of Mathematics
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• v.21 no.2
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• pp.151-159
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• 2013

An $L(j,k)$-labeling of a graph G is a vertex labeling such that the difference of the labels of any adjacent vertices is at least $j$ and that of any vertices of distance two is at least $k$ for given $j$ and $k$. The minimum span of all L(2, 1)-labelings of G is called the ${\lambda}$-number of G and is denoted by ${\lambda}(G)$. In this paper, we find a lower bound of the ${\lambda}$-number of the Cartesian product $K_m{\Box}C_n$ of the complete graph $K_m$ of order $m$ and the cycle $C_n$ of order $n$. In fact, we show that when $n{\geq}3$, ${\lambda}(K_4{\Box}C_n){\geq}7$ and the equality holds if and only if n is a multiple of 8. Moreover when $m{\geq}5$, ${\lambda}(K_m{\Box}C_n){\geq}2m-1$ and the equality holds if and only if $n$ is even.