1. L(4, 3, 2, 1)-Labeling for Simple GraphsSoumen Atta, Priya Ranjan Sinha Mahapatra, 2015, published scientific conference contribution Abstract: An L(4, 3, 2, 1)-labeling of a graph is a function which assigns label to each vertex of the graph such that if two vertices are one, two, three and four distance apart then assigned labels must have a difference of at least 4, 3, 2 and 1 respectively between them. This paper presents L(4, 3, 2, 1)-labeling number for simple graphs such as complete graphs, complete bipartite graphs, stars, paths and cycles. This paper also presents an L(4, 3, 2, 1)-labeling algorithm for paths which is optimal for paths on n≥7 vertices. Keywords: L(4, 3, 2, 1)-labeling, Labeling number, Graph labeling, Channel assignment problem Published in RUNG: 05.06.2023; Views: 1390; Downloads: 0 This document has many files! More... |
2. L (D, 2, 1)-labeling of Square GridSoumen Atta, Priya Ranjan Sinha Mahapatra, 2019, original scientific article Abstract: For a fixed integer $D (\geq 3)$ and $\lambda$ $\in$ $\mathbb{Z}^+$, a $\lambda$-$L(D, 2, 1)$-$labeling$ of a graph $G = (V, E)$ is the problem of assigning non-negative integers (known as labels) from the set $\{0, \ldots, \lambda\}$ to the vertices of $G$ such that if any two vertices in $V$ are one, two and three distance apart from each other then the assigned labels to these vertices must have a difference of at least $D$, 2 and 1 respectively. The vertices which are at least $4$ distance apart can receive the same label. The minimum value among all the possible values of $\lambda$ for which there exists a $\lambda$-$L(D, 2, 1)$-$labeling$ is known as the labeling number. In this paper $\lambda$-$L(D, 2 ,1)$-$labeling$ of square grid is considered. The lower bound on the labeling number for square grid is presented and a formula for $\lambda$-$L(D, 2 ,1)$-$labeling$ of square grid is proposed. The correctness proof of the proposed formula is given here. The upper bound of the labeling number obtained from the proposed labeling formula for square grid matches exactly with the lower bound of the labeling number. Keywords: Graph labeling, Square grid, Labeling number, Frequency assignment problem (FAP) Published in RUNG: 17.04.2023; Views: 1410; Downloads: 0 This document has many files! More... |
3. No-hole λ-L (k, k – 1, …, 2,1)-labeling for square gridSoumen Atta, Stanisław Goldstein, Priya Ranjan Sinha Mahapatra, 2017, original scientific article Abstract: Motivated by a frequency assignment problem, we demonstrate, for a fixed positive integer k, how to label an infinite square grid with a possibly small number of integer labels, ranging from 0 to λ −1, in such a way that labels of adjacent vertices differ by at least k, vertices connected by a path of length two receive values which differ by at least k − 1, and so on. The vertices which are at least k + 1 distance apart may receive the same label. By finding a lower bound for λ, we prove that the solution is close to optimal, with approximation ratio at most 9/8. The labeling presented is a no-hole one, i.e., it uses each of the allowed labels at least once. Keywords: graph labeling, labeling number, no-hole labeling, square grid, frequency assignment problem, approximation ratio Published in RUNG: 17.04.2023; Views: 1841; Downloads: 0 This document has many files! More... |