Elsevier

Discrete Applied Mathematics

Volume 291, 11 March 2021, Pages 264-270
Discrete Applied Mathematics

L(p,q)-labelings of subdivisions of graphs

https://doi.org/10.1016/j.dam.2020.12.019Get rights and content

Abstract

Given a graph G and a function h from E(G) to N, the h-subdivision of G, denoted by G(h), is the graph obtained from G by replacing each edge uv in G with a path P:uxuv1xuv2xuvn1v, where n=h(uv). When h(e)=c is a constant for all eE(G), we use G(c) to replace G(h). For a given graph G, an L(p,q)-labeling of G is a function f from the vertex set V(G) to the set of all nonnegative integers such that f(u)f(v)p if dG(u,v)=1, and f(u)f(v)q if dG(u,v)=2. A k-L(p,q)-labeling is an L(p,q)-labeling such that no label is greater than k. The L(p,q)-labeling number of G, denoted by λp,q(G), is the smallest number k such that G has a k-L(p,q) -labeling. We study the L(p,q)-labeling numbers of subdivisions of graphs in this paper. We prove that λp,q(G(3))=p+(Δ1)q when p2q and Δ>2pq, and show that λp,q(G(h))=p+(Δ1)q when p2q and Δ3pq, where h is a function from E(G) to N so that h(e)3 for all eE(G).

Introduction

In the channel assignment problem, we need to assign frequency bands to transmitters, if two transmitters are too close, interference will occur if they attempt to transmit on close frequencies. In order to avoid this situation, the separation of the channels assigned to them must sufficient. Moreover, if two transmitters are close but not too close, the channel assigned must be different. This problem was known under the L(p,q)-labeling problem of a graph G, where an L(p,q)-labeling of G is a nonnegative integer assignment f to the vertices of G such that for all u,v in V(G), dG(u,v)=1 implies |f(u)f(v)|p, and dG(u,v)=2 implies |f(u)f(v)|q. A k-L(p,q)-labeling is an L(p,q)-labeling such that no label is greater than k. The L(p,q)-labeling number of G, denoted by λp,q(G) (λd,1(G)=λd(G), and λ2,1(G)=λ(G)), is the smallest number k such that G has a k-L(p,q)-labeling. This problem was first proposed by Griggs and Yeh [10] and has been studied extensively over the past years. Griggs and Yeh [10] showed that the L(2,1)-labeling problem is NP-complete for general graphs and proved that λ(G)Δ2(G)+2Δ(G), where Δ(G) is the maximum degree of a vertex in G. Chang and Kuo [7] gave a polynomial-time algorithm for the L(2,1)-labeling problem on trees. Gon çalves [9] proved that λ(G)Δ2(G)+Δ(G)2 for any graph G with Δ(G)2. There are also many other results around this topic, for an overview on the subject, we refer the reader to the surveys in [5] and [23].

Whittlesty, Georges and Mauro [21] studied the L(2,1)-labeling of the incidence graph of G, which is the graph obtained from G by replacing each edge by a path of length 2. The L(2,1)-labeling of the incidence graph of G is equivalent to an assignment of integers to each element of V(G)E(G) of G such that: (i) any two adjacent vertices of G receive different integers; (ii) any two adjacent edges of G receive different integers; and (iii) a vertex and its incident edge receive integers that differ by at least 2 in absolute value. This labeling is called (2,1)-total labeling of G, which was introduced by Havet and Yu in 2002 [11], [12] and generalized to the (d,1) -total labeling of a graph G. In [12], Havet and Yu obtained the bound Δ(G)+d1λdT(G)2Δ(G)+d1 and conjectured λdT(G)Δ(G)+2d1 (λdT(G) is the (d,1)-total labeling number of G). It is sufficient to prove the conjecture for Δ(G)>d. Havet and Yu [12] completed the proof of the conjecture for Δ(G)3 by proving that λ2T(G)6 if Δ(G)3. Note that if d=1, then the (1,1)-total labeling is the traditional total coloring. The total coloring and (d,1)-total labeling has been intensively studied in [1], [2], [3], [4], [8], [13], [17], [18], [19], [20], [22].

Given a graph G and a function h from E(G) to N, the h-subdivision of G, denoted by G(h), is the graph obtained from G by replacing each edge uv in G with a path P:u(xuv1)h(xuv2)h(xuvn1)hv (or simply, uxuv1xuv2xuvn1v, if the function h need not to be specified), where n=h(uv). If h is a constant function; that is, h(e)=c for all eE(G), where c is a constant, we use G(c) to replace G(h). And we use xuvi to replace (xuvi)h when h(e)=c for all eE(G) for simplicity.

Lű [15] studied the L(2,1)-labeling problem of subdivisions of graphs. This problem can be viewed as a generalization of the (2,1)-total labeling problem. Note that λ(G(1))=λ(G) and λ(G(2))=λ2T(G) for any graph G. In [15], Lű proved that for all c4, λ(G(c))Δ(G)+2 for any graph G. He also showed that λ(G(3))Δ(G)+4 for any graph G. Based on these, he conjectured that λ(G(3))Δ(G)+2 for any graph G. Lű and Lin [16] studied the L(d,1)-labeling problem of subdivisions of graphs and showed that λd(G(3))Δ(G)+2d1 for any graph G with Δ(G)3. From this, they deduced that λ(G(3))Δ(G)+3 for any graph G. Karst et al. [14] showed that λd(G(3))d+Δ(G)2+max{Δ(G)2,d}1 for any graph G with Δ(G)3. From this, they deduced that the conjecture proposed by Lű [15] is true, and, as a bonus, they obtained that λ(G(3))=Δ(G)+1 for any graph G with even Δ4. Chang et al. [6] studied the L(2,1)-labeling numbers of subdivisions of graphs, they proved that λ(G(3))=Δ(G)+1 for any graph G with Δ(G)4. and that λ(G(h))=Δ(G)+1 if Δ(G)5 and h is a function from E(G) to N so that h(e)3 for all eE(G), or if Δ(G)4 and h is a function from E(G) to N so that h(e)4 for all eE(G).

We study the L(p,q)-labeling numbers of subdivisions of graphs in this paper. We extend the results given in [6] and prove that λp,q(G(3))=p+(Δ1)q when p2q and Δ>2pq. Based on this, we deduce that λp,q(G(h))=p+(Δ1)q when p2q and Δ3pq, where h is a function from E(G) to N so that h(e)3 for all eE(G).

Section snippets

Preliminaries

Throughout this paper, we assume that pq. We give some basic properties for the L(p,q)-labeling number of graphs in this section. Given a graph G, a vertex v in G is called major if degG(v)=Δ(G), where degG(v) is the degree of v in G. The following two lemmas are easy to verify.

Lemma 1

If G is a graph of maximum degree Δ(G)=Δ1, then λp,q(G)p+(Δ1)q. Moreover, if λ(G)=p+(Δ1)q and f is a (p+(Δ1)q)-L(p,q)-labeling of G, then f(v)=0 or p+(Δ1)q for every major vertex v in G.

Lemma 2

If f is a k-L(p,q)-labeling

L(p,q)-labeling number of G(3)

We study the L(p,q)-labeling number of G(3) in this section. We first give some definitions and fix some notations that will be used later. Given a graph G, a u-v walk of length n in G is a sequence of vertices and edges of the form u0,e1,u1,e2,,en,un, where u=u0, v=un, ei=ui1ui for i=1,2,,n. Since the sequence of vertices of a walk determines the walk, we often use u0u1...un to denote a walk. A u-v trail is a walk in which all edges are distinct. For a vertex v in V(G), the neighborhood of v

L(p,q)-labeling number of G(h)

We now consider the L(p,q)-labeling number of G(h). From now on, for convenience, for two integers a, b and a set of integers S, we use the symbol BS(a;b) to denote the set {c:cS, |ca|<b}.

Given a graph G together with an edge uv in E(G), we can construct a graph G with V(G)=V(G){z}, E(G)=(E(G){uz,zv}){uv}. If h is a function from E(G) to N with h(uv)=l2, then for each i with 1il1, the function hi defined on E(G), which is defined by hi(e)=h(e),if eE(G){uv},i,if e=uz,li,if e=zv,

Concluding notes

In this paper, we showed that if p2q, then λp,q(G(h))=p+(Δ1)q, when Δ>2pq and h is the function from E(G) to N so that h(e)=3 for all eE(G), or when Δ3pq and h is a function from E(G) to N so that h(e)3 for all eE(G). We believe that a more general result, which covers both Theorem 5, Theorem 7, is true. In fact, we have the following conjecture.

Conjecture 8

Let G be a graph with maximum degree Δ, and p,qN with p2q, then λp,qG3=p+Δ1q whenever Δ2pq, and h is a function from E(G) to N so that h(e)3

Acknowledgments

The authors would like to express their gratitude to the referees for their valuable comments and suggestions on improving the presentation of this paper.

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  • Cited by (1)

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    Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST107-2115-M-003-005-MY2.

    2

    Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST106-2115-M-156-002.

    3

    Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST104-2115-M-259-002-MY2.

    4

    Supported in part by the Ministry of Science and Technology of Taiwan under Grant No. MOST106-2115-M-008-012-MY2.

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