List graphs and distance-consistent node labelings

In this note we present a few open problems on various aspects of graph labelings, which have not been included in any of the other papers appearing in this volume.

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, 2015

We consider how to assign labels to any undirected graph with n nodes such that, given the labels of two nodes and no other information regarding the graph, it is possible to determine the distance between the two nodes. The challenge in such a distance labeling scheme is primarily to minimize the maximum label lenght and secondarily to minimize the time needed to answer distance queries (decoding). Previous schemes have offered different trade-offs between label lengths and query time. This paper presents a simple algorithm with shorter labels and shorter query time than any previous solution, thereby improving the state-of-the-art with respect to both label length and query time in one single algorithm. Our solution addresses several open problems concerning label length and decoding time and is the first improvement of label length for more than three decades. More specifically, we present a distance labeling scheme with labels of length log 3 2 n + o(n) bits 1 and constant decoding time. This outperforms all existing results with respect to both size and decoding time, including Winkler's (Combinatorica 1983) decade-old result, which uses labels of size (log 3)n and O(n/ log n) decoding time, and Gavoille et al. (SODA'01), which uses labels of size 11n + o(n) and O(log log n) decoding time. In addition, our algorithm is simpler than the previous ones. In the case of integral edge weights of size at most W , we present almost matching upper and lower bounds for the label size ℓ: . Furthermore, for r-additive approximation labeling schemes, where distances can be off by up to an additive constant r, we present both upper and lower bounds. In particular, we present an upper bound for 1-additive approximation schemes which, in the unweighted case, has the same size (ignoring second order terms) as an adjacency labeling scheme, namely n/2. We also give results for bipartite graphs as well as for exact and 1-additive distance oracles.

Certain results on prime and prime distance labeling of graphs, 2020

Let G be a graph on n vertices. A bijective function f : V (G) → {1, 2, ..., n} is said to be a prime labeling if for every e = xy, GCD{f (x), f (y)} = 1. A graph which permits a prime labeling is a "prime graph". On the other hand, a graph G is a prime distance graph if there is an injective function g : V (G) → Z (the set of all integers) so that for any two vertices s & t which are adjacent, the integer |g(s) − g(t)| is a prime number and g is called a prime distance labeling of G. A graph G is a prime distance graph (PDL) iff there exists a "prime distance labeling" (PDL) of G. In this paper, we obtain the prime labeling and prime distance labeling of certain classes of graphs.

Electronic Journal of Graph Theory and Applications, 2021

Let G be a graph with |V (G)| vertices and ψ : V (G) −→ {1, 2, 3, • • • , |V (G)|} be a bijective function. The weight of a vertex v ∈ V (G) under ψ is w ψ (v) = u∈N (v) ψ(u). The function ψ is called a distance magic labeling of G, if w ψ (v) is a constant for every v ∈ V (G). The function ψ is called an (a, d)-distance antimagic labeling of G, if the set of vertex weights is a, a + d, a + 2d,. .. , a + (|V (G)| − 1)d. A graph that admits a distance magic (resp. an (a, d)distance antimagic) labeling is called distance magic (resp. (a, d)-distance antimagic). In this paper, we characterize distance magic 2-regular graphs and (a, d)-distance antimagic some classes of 2-regular graphs.

2019

Stimulated by the famous plane coloring problem Eggleton coined the term distance graph and studied widely the prime distance graphs. A prime distance graph (PDG) G(Z,D) is one whose vertex set V is the set of integers Z and the distance set D is a subset of the set of primes P . The edge set of G denoted E is the one whose elements (u, v) for any u, v ∈ V (G) are characterized by the property that d(u, v) ∈ D where d(u, v) = |u − v| . According to J.D.Laison, C. Starr and A. Walker a graph G is a PDG if there exists a 1-1 labeling f : V (G)→ Z such that for any two adjacent vertices u and v the integer |f(u)− f(v)| is a prime. Further they called such a labelling of V (G) a prime distance labelling (PDL) of G . In this paper we prove certain existence and non-existence results concerning PDG and PDL and study the relationship between them. We also discuss certain applications besides raising some open problems.

2018

A graph G is a prime distance graph if there exists an one-to-one labeling of its vertices f : V (G) → Z such that for any two adjacent vertices u and v in G, the integer |f(u) − f(v)| is a prime. So G is a prime distance graph if and only if there exists a prime distance labeling of G. In this paper we derive certain interesting results concerning prime distance labeling of some path related graphs in the context of some graph operations, namely, power, duplication, extension and vertex switching in a path Pn. AMS Subject Classification: 05C15

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.

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 Mathematics, 2001

A k-circular distance two labelling (or k-c-labelling) of a graph G is a vertex-labelling such that the circular di erence (mod k) of the labels is at least two for adjacent vertices, and at least one for vertices at distance two. Given G, denote (G) the minimum k for which there exists a k-c-labelling of G. Suppose G has n vertices, we prove (G)6n if G c is Hamiltonian; and (G) = n + pv(G c) otherwise, where pv(G) is the path covering number of G. We give exact values of (G) for some families of graphs such that G c is Hamiltonian, and discuss injective k-c-labellings especially for joins and unions of graphs.

Theoretical Computer Science, 2005

Let G = (V , E) be a simple graph, and for all v ∈ V , let L(v) be a list of colors assigned to v. We shall assume throughout that the colors are natural numbers. For nonnegative integers d, s, define d,s (G) to be the smallest integer k such that for every list assignment with |L(v)| = k for all v ∈ V one can choose a color c(v) ∈ L(v) for every vertex in such a way that |c(v) − c(w)| d for all vw ∈ E and |c(v) − c(w)| s for all pairs v, w of vertices having distance 2 in G. For a given list assignment such a coloring c is called an L(d, s)-list labeling.