Labeling Schemes for Bounded Degree Graphs

Discrete Mathematics, 2000

Given a graph G and a positive integer d, an L(d; 1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u) − f(v)|¿d; if u and v are not adjacent but there is a two-edge path between them, then

Discrete Applied Mathematics, 2011

Motivated by applications in software programming, we consider the problem of covering a graph by a feasible labeling. Given an undirected graph G = (V , E), two positive integers k and t, and an alphabet Σ, a feasible labeling is defined as an assignment of a set L v ⊆ Σ to each vertex v ∈ V , such that (i) |L v | ≤ k for all v ∈ V and (ii) each label α ∈ Σ is used no more than t times. An edge e = {i, j} is said to be covered by a feasible labeling if L i ∩ L j ̸ = ∅. G is said to be covered if there exists a feasible labeling that covers each edge e ∈ E. In general, we show that the problem of deciding whether or not a tree can be covered is strongly NP-complete. For k = 2, t = 3, we characterize the trees that can be covered and provide a linear time algorithm for solving the decision problem. For fixed t, we present a strongly polynomial algorithm that solves the decision problem; if a tree can be covered, then a corresponding feasible labeling can be obtained in time polynomial in k and the size of the tree. For general graphs, we give a strongly polynomial algorithm to resolve the covering problem for k = 2, t = 3.

2017

A positive integern is called super totient if the residues of n which are prime ton can be partitioned into two disjoint subsets of equal sums. LetG be a given graph withV, the set of vertices and E is the set of its edges. An injective function g defined onV into subset of integers will be termed as super totient labeling of the graph G, if the functiong∗ : E → N defined byg∗(xy) = g(x)g(y) assigns a super totient number for all edgesxy ∈ E, wherex, y ∈ V. A graph admits this labeling is called a super totient graph. In the current manuscript, the authors investigate a novel labeling algorithm, called super totient labeling, for several classes of graphs such as friendship graphs, wheel graphs, complete graphs and complete bipartite graphs. AMS (MOS) Mathematics subject classification (2000): 05C25, 11E04, 20G15

Discrete Mathematics, 2000

Given a graph G and a positive integer d, an L(d; 1)-labeling of G is a function f that assigns to each vertex of G a non-negative integer such that if two vertices u and v are adjacent, then |f(u) − f(v)|¿d; if u and v are not adjacent but there is a two-edge path between them, then |f(u) − f(v)|¿1. The L(d; 1)-number of G, d (G), is deÿned as the minimum m such that there is an L(d; 1)-labeling f of G with f(V) ⊆{0; 1; 2; : : : ; m}. Motivated by the channel assignment problem introduced by Hale (Proc. IEEE 68 (1980) 1497-1514), the L(2; 1)-labeling and the L(1; 1)-labeling (as d = 2 and 1, respectively) have been studied extensively in the past decade. This article extends the study to all positive integers d. We prove that d (G)6 2 + (d − 1) for any graph G with maximum degree. Di erent lower and upper bounds of d (G) for some families of graphs including trees and chordal graphs are presented. In particular, we show that the lower and the upper bounds for trees are both attainable, and the upper bound for chordal graphs can be improved for several subclasses of chordal graphs.

Theoretical Computer Science, 2021

A radio labeling of a graph G is a mapping f : V (G) → {0, 1, 2,...} such that |f (u) − f (v)| d(G) + 1 − d(u, v) holds for every pair of vertices u and v, where d(G) is the diameter of G and d(u, v) is the distance between u and v in G. The radio number of G, denoted by rn(G), is the smallest t such that G admits a radio labeling with t = max{|f (v) − f (u)| : v, u ∈ V (G)}. A block graph is a graph such that each block (induced maximal 2-connected subgraph) is a complete graph. In this paper, a lower bound for the radio number of block graphs is established. The block graph which achieves this bound is called a lower bound block graph. We prove three necessary and sufficient conditions for lower bound block graphs. Moreover, we give three sufficient conditions for a graph to be a lower bound block graph. Applying the established bound and conditions, we show that several families of block graphs are lower bound block graphs, including the level-wise regular block graphs and the extended star of blocks. The line graph of a graph G(V, E) has E(G) as the vertex set, where two vertices are adjacent if they are incident edges in G. We extend our results to trees as trees and its line graphs are block graphs. We prove that if a tree is a lower bound block graph then, under certain conditions, its line graph is also a lower bound block graph, and vice versa. Consequently, we show that the line graphs of many known lower bound trees, excluding paths, are lower bound block graphs.

Discrete Applied Mathematics, 2013

An L(2, 1)-labeling of a graph G is an assignment of a nonnegative integer to each vertex of G such that adjacent vertices receive integers that differ by at least two and vertices at distance two receive distinct integers. The span of such a labeling is the difference between the largest and smallest integers used. The λ-number of G, denoted by λ(G), is the minimum span over all L(2, 1)-labelings of G. Bodlaender et al. conjectured that if G is an outerplanar graph of maximum degree ∆, then λ(G) ≤ ∆ + 2. Calamoneri and Petreschi proved that this conjecture is true when ∆ ≥ 8 but false when ∆ = 3. Meanwhile, they proved that λ(G) ≤ ∆ + 5 for any outerplanar graph G with ∆ = 3 and asked whether or not this bound is sharp. In this paper we answer this question by proving that λ(G) ≤ ∆ + 3 for every outerplanar graph with maximum degree ∆ = 3. We also show that this bound ∆ + 3 can be achieved by infinitely many outerplanar graphs with ∆ = 3.

Discrete Applied Mathematics, 2010

A mapping from the vertex set of a graph G = (V , E) into an interval of integers {0, . . . , k} is an L(2, 1)-labelling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices at distance 2 are mapped onto distinct integers. It is known that, for any fixed k ≥ 4, deciding the existence of such a labelling is an NP-complete problem while it is polynomial for k ≤ 3. For even k ≥ 8, it remains NP-complete when restricted to planar graphs. In this paper, we show that it remains NP-complete for any k ≥ 4 by reduction from Planar Cubic Two-Colourable Perfect

Theoretical Computer Science, 1982

Given a tree T with n edges and a set W of n weights, we deal with labelings of the edges of T with weights from W, optimizing certain objective functiorls. For some of these function& the optimization problem is shown to be NP-complete (e.g., finding a labeling with minimai diameter), and for others we find polynomial-time algorithms (e.g., finding a labeling with minimal average distance).

Mathematics and Statistics, 2022

An assignment of intergers to the vertices of a graphḠ subject to certain constraints is called a vertex labeling ofḠ. Different types of graph labeling techniques are used in the field of coding theory, cryptography, radar, missile guidance, x-ray crystallography etc. A DCL ofḠ is a bijective functionf from node setV ofḠ to {1, 2, 3, ..., |V |} such that for each edge rs, we allot 1 iff (r) dividesf (s) or f (s) dividesf (r) \u0026 0 otherwise, then the absolute difference between the number of edges having 1 \u0026 the number of edges having 0 do not exceed 1, i.e., |ef (0) − ef (1)| ≤ 1. IfḠ permits a DCL, then it is called a DCG. A complete graph K n , is a graph on n nodes in which any 2 nodes are adjacent and lilly graph I n is formed by 2K 1,n joining 2P n , n ≥ 2 sharing a common node. i.e., I n = 2K 1,n + 2P n , where K 1,n is a complete bipartite graph \u0026 P n is a path on n nodes. In this paper, we propose an interesting conjecture concerning DCL for a givenḠ, besides, discussing certain general results concerning DCL of complete graph K n −related graphs. We also prove that I n admits a DCL for all n ≥ 2. Further, we establish the DCL of some I n −related graphs in the context of some graph operations such as duplication of a node by an edge, node by a node, extension of a node by a node, switching of a node, degree splitting graph, \u0026 barycentric subdivision of the givenḠ.

Lecture Notes in Computer Science, 2014

We investigate labeling schemes supporting adjacency, ancestry, sibling,and connectivity queries in forests. In the course of more than 20 years, the existence of log n + O(log log n) labeling schemes supporting each of these functions was proven, with the most recent being ancestry [Fraigniaud and Korman, STOC '10]. Several multi-functional labeling schemes also enjoy lower or upper bounds of log n + Ω(log log n) or log n + O(log log n) respectively. Notably an upper bound of log n + 2 log log n for adjacency+siblings and a lower bound of log n + log log n for each of the functions siblings, ancestry, and connectivity [Alstrup et al., SODA '03]. We improve the constants hidden in the O-notation, where our main technical contribution is a log n + 2 log log n lower bound for connectivity+ancestry and connectivity+siblings. In the context of dynamic labeling schemes it is known that ancestry requires Ω(n) bits [Cohen, et al. PODS '02]. In contrast, we show upper and lower bounds on the label size for adjacency, siblings, and connectivity of 2 log n bits, and 3 log n to support all three functions. We also show that there exist no efficient dynamic adjacency labeling schemes for planar, bounded treewidth, bounded arboricity and bounded degree graphs.

A simple undirected graph G is called a sum graph if there exists a labeling f of the vertices of G into distinct positive integers such that any two distinct vertices u and v of G are adjacent if and only if there is a vertex w in G with label f (w) = f (u) + f (v). In this paper we give an algorithm for labeling a graph G of order n and size m when its adjacency matrix is given. We prove that The maximum integer in the sum labeling is bounded from above by 2 n 2 .3 n 2. Also we give an algorithm for optimal sum labeling of trees.

Anais do VI Encontro de Teoria da Computação (ETC 2021), 2021

We compare the behaviour of the $L(h,k)$-number of undirected and oriented graphs in terms of maximum degree, highlighting differences between the two contexts. In particular, we prove that, for every $h$ and $k$, oriented graphs with bounded degree in every block of their underlying graph (for instance, oriented trees and oriented cacti) have bounded $L(h,k)$-number, giving an upper bound on this number which is sharp up to a multiplicative factor $4$.

Malaya Journal of Matematik

A labeling of a graph is a mapping that maps some set of graph elements to a set of numbers. In this paper, two new variations of labeling named k-distant edge total labeling and k-distant vertex total labeling are introduced. Moreover, the study of two new graph parameters, called k-distant edge chromatic number (γ kd) and k-distant vertex chromatic number (γ kd) related this labeling are initiated. The k-distant vertex total labeling for paths, cycles, complete graphs, stars, bi-stars and friendship graphs are studied and the value of the parameter γ kd determined for these graph classes. Then k-distant edge total labeling for paths, cycles and stars are studied. Also, an upper bound of γ kd and a lower bound of γ kd are presented for general graphs.

Discrete Mathematics, 2012

Let G be a graph with vertex set V (G) and edge set E(G), and f be a 0 − 1 labeling of E(G) so that the absolute difference in the number of edges labeled 1 and 0 is no more than one. Call such a labeling f edge-friendly. We say an edge-friendly labeling induces a partial vertex labeling if vertices which are incident to more edges labeled 1 than 0, are labeled 1, and vertices which are incident to more edges labeled 0 than 1, are labeled 0. Vertices that are incident to an equal number of edges of both labels we call unlabeled. Call a procedure on a labeled graph a label switching algorithm if it consists of pairwise switches of labels. Given an edge-friendly labeling of K n , we show a label switching algorithm producing an edge-friendly relabeling of K n such that all the vertices are labeled. We call such a labeling opinionated.

Proceedings of the 29th International Conference on Scientific and Statistical Database Management, 2017

Given a directed graph, how should we label both its outgoing and incoming edges to achieve be er disk locality and support neighborhood-related edge queries? In this paper, we answer this question with edge labeling schemes G R and F I O , to label edges with integers based on the premise that edges should be assigned integer identi ers exploiting their consecutiveness to a maximum degree. We provide extensive experimental analysis on real-world graphs, and compare our proposed schemes with other labeling methods based on assigning edge IDs in the order of insertion or even randomly, as traditionally done. We show that our methods are e cient and result in signi cantly improved query I/O performance, including with indexes built on directed a ributed edges. is ultimately leads to faster execution of neighborhood-related edge queries.