Distance-two labelings of graphs

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www.elsevier.com/locate/ejc

Distance-two labelings of graphs

Gerard J. Chang

a

, Changhong Lu

b,1 a_{Department of Mathematics, National Taiwan University, Taipei 106, Taiwan}

b_{Department of Mathematics, Hunan Normal University, Changsha 410081, People’s Republic of China}

Received 25 May 2001; accepted 26 September 2002

Abstract

For given positive integers j ≥ k, an L( j, k)-labeling of a graph G is a function f : V (G) → {0, 1, 2, . . .} such that | f (u) − f (v)| ≥ j when dG(u, v) = 1 and | f (u) − f (v)| ≥ k when dG(u, v) = 2. The L( j, k)-labeling number λj,k(G) of G is defined as the minimum m such that there is an L( j, k)-labeling f of G with f (V (G)) ⊆ {0, 1, 2, . . . , m}. For a graph G of maximum degree∆ ≥ 1 it is the case that λ_j_,k(G) ≥ j + (∆ − 1)k. The purpose of this paper is to study the structures of graphs G with maximum degree∆ ≥ 1 and λj_,k(G) = j + (∆ − 1)k.

c

Keywords: Distance; Labeling; Degree; Tree; Star; Algorithm; Neighbor

1. Introduction

The problem of vertex labeling with a condition at distance two, proposed by Griggs and Roberts [16], arose from a variation of the channel assignment problem introduced by Hale[10]. Suppose a number of transmitters are given. We must assign a channel to each of the given transmitters such that the interference is avoided. In order to reduce the interference, any two “close” transmitters must receive channels at least k apart, and any two “very close” transmitters must receive channels at least j apart, where j ≥ k are two given positive integers. One can construct an interference graph for this problem so that the transmitters are the vertices and there is an edge joining two “very close” transmitters. Two transmitters are defined as “close” if the corresponding vertices are of distance two.

Then, for a given graph G, an L( j, k)-labeling is defined as a function f : V (G) → {0, 1, 2, . . .} such that | f (u) − f (v)| ≥ j when dG(u, v) = 1 and | f (u) − f (v)| ≥ k when dG(u, v) = 2, where dG(u, v), the distance of u and v, is the minimum length of a path between u andv. The L( j, k)-labeling number λj,k(G) of G is the smallest number

1 Permanent address: Department of Mathematics, East China Normal University, Shanghai 200062, P. R. China.

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λj, j(T ) = j∆. For j = 2 and k = 1, Griggs and Yeh

means of a first-fit (greedy) algorithm: for any tree T with maximum vertex degree∆ ≥ 1,

∆ + 1 ≤ λ2,1(T ) ≤ ∆ + 2.

This result gives rise to the question of classifying those trees withλ2,1(T ) = ∆ + 1 or alternatively∆ + 2. Although Griggs and Yeh conjectured that the classification problem is NP-complete, Chang and Kuo[2]presented a polynomial–time classification algorithm. The same procedure was also carried out for the L( j, 1)-labelings. It was proved by Chang et al.[1]that for any tree of maximum degree∆,

∆ + j − 1 ≤ λj,1(T ) ≤ min{∆ + 2 j − 2, 2∆ + j − 2}.

Moreover, the lower bound and the upper bounds are both attainable. They also gave a polynomial–time algorithm for determiningλj,1(T ) of a tree T .

We now consider the lower bound on the above two cases for general graphs G and general positive integers j≥ k.

Proposition 1. For any positive integers j ≥ k and any graph G of maximum degree

∆ ≥ 1, we have λj,k(G) ≥ j + (∆ − 1)k. Moreover, if j > k and the equality holds, then

for anyλj,k-labeling of G each major vertex (i.e. a vertex of degree∆) must be labeled 0

(or j+ (∆ − 1)k) and its neighbors must be labeled j + ik (or ik) for i = 0, 1, . . . , ∆ − 1.

We call a graph G of maximum degree∆ ≥ 1 with λj,k(G) = j + (∆ − 1)k a λj,k

-minimal graph. It is clear that K1,∆ isλj,k-minimal for all∆ ≥ 1 and j ≥ k; see[5]. While determiningλj,k(G) for a graph is in general not an easy job, the aim of this paper is to study the structures ofλj,k-minimal graphs.

2. λλλjjj,k,k,k-minimal graphs

We first study the structures ofλj,k-minimal graphs.

Notice that whether a graph isλj,k-minimal or not depends on the values of j and k. As an example, for∆ ≥ 1 consider the double star D_∆ which is obtained from two stars

K1,∆by identifying a leaf of one star with a leaf of the other star.Fig. 1shows the double star D6.

ByProposition 1, if D_∆ isλj,k-minimal then one of the two major vertices is labeled by 0 and its neighbors by j, j +k, . . . , j +(∆−1)k, and the other is labeled by j +(∆−1)k and its neighbors by 0, k, . . . , (∆−1)k. Hence, the only common neighbor of the two major vertices is labeled by j+ i k = i k for some 0≤ i , i ≤ ∆ − 1. This is possible if and

only if j = ik for some integer 1 ≤ i ≤ ∆ − 1. In conclusion, D_∆isλj,k-minimal if and only if j = ik for some integer 1 ≤ i ≤ ∆ − 1.

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Fig. 1. The double star D6.

For some cases, λj,k-minimality does imply other λp,q-minimality. Consider the following result in[5].

Lemma 2. The following are true for any graph G.

(1) λcj,ck(G) = cλj,kfor all positive integers c and j ≥ k.

(2) λj,k(G) ≤ λp,q(G) for all positive integers j ≥ k and p ≥ q satisfying j ≤ p and

k≤ q.

We then have

Theorem 3. Suppose G is a graph of maximum degree∆ ≥ 1; and j ≥ k and p ≥ q are

positive integers satisfying p= q j/k. If G is λj,k-minimal, then G isλp,q-minimal.

Proof. According toLemma 2, we have

kλp,q(G) = λkp,kq ≤ λq j,qk = qλj,k= q( j + (∆ − 1)k).

Hence, λp,q(G) ≤ q j/k + (∆ − 1)q implying λp,q(G) ≤ p + (∆ − 1)q and so

λp,q(G) = p + (∆ − 1)q. This completes the proof of the theorem.

As seen above, not only is determiningλj,k(G) hard, but also deciding whether G is

λj,k-minimal may be not easy as it depends on both the structure of G and the values of j and k. In the rest of the paper, we restrict our attention to graphs that areλp,q-minimal for all p≥ q.

We then have the following theorem.

Theorem 4. Suppose G is a graph of maximum degree∆ ≥ 1 and j ≥ k are positive

integers satisfying j≥ ∆k. Then the following statements are equivalent.

(1) G isλj,k-minimal.

(2) G has aλj,k-labeling g such that any vertexv in G, g(v) is of the form avj+ bvk

with a_v ∈ {0, 1} and bv ∈ {0, 1, . . . , ∆ − 1}. Moreover, the following statements hold.

(D1) If dG(u, v) = 1, then au= av. If au= 0 and av= 1, then bu≤ bv. (D2) If dG(u, v) = 2, then au= avand bu= bv.

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(v) < f (u) and so f (v) ≤ f (u) − j ≤ (∆ − 1)k implying bv ≤ ∆ − 1. If such u does not exist, then we may assume that f(v) = 0. In any case, we have 0 ≤ b_v≤ ∆ − 1.

Now, consider the function g on V(G) defined by g(v) = a_vj+ b_vk for all verticesv

in G. We shall check that g is an L( j, k)-labeling of G as desired.

f(u) < f (v) and so in fact f (u) ≤ f (v) − j implying bu≤ bv. Then g(u) ≤ g(v) + j. Next, suppose dG(u, v) = 2. Choose a vertex w adjacent to both u and v. Then, au= aw and a_w = a_vimplying that au= av. So, we have k ≤ | f (u)− f (v)| = |buk+ru−bvk−rv| which implies bu= bv. Then|g(u) − g(v)| ≥ k.

(2) ⇒ (3). Define a labeling h : V (G) → {0, 1, 2, . . .} by h(v) = avp + b_vq

for all vertices v in G. If dG(u, v) = 1, then (D1) implies |h(u) − h(v)| ≥ p; if

dG(u, v) = 2, then (D2) implies |h(u) − h(v)| ≥ q. Hence, h is an L(p, q)-labeling

of G with max_{v∈V (G)}h(v) ≤ p + (∆ − 1)q. Thus, G is λp,q-minimal.

(3) ⇒ (1). This follows by taking p = j and q = k.

Corollary 5. If G is aλj,k-minimal graph of maximum degree∆ ≥ 1 and j ≥ ∆k, then

G is bipartite.

Proof. Notice that inTheorem 4, every vertex has a label of the form a_vj+ b_vk uniquely.

According to condition (D1), au = avwhenever uv is an edge. Thus, G is a bipartite graph with a bipartition of V(G) into A = {v : a_v= 0} and B = {v : a_v= 1}.

It is desirable to design an efficient algorithm to decide whether a graph isλp,q-minimal for all positive integers p≥ q. We are only able to do this for trees. This will be discussed in the next section.

3. λλλjjj,k,k,k-minimal trees

We now turn our attention to the case where the graph is a tree. For this case, an alternative statement ofTheorem 4is as follows.

For any positive integer∆, a ∆-sequence is a sequence (b0, b1, . . . , bm) of integers satisfying (S1) to (S4).

(S1) b0= 0.

(S2) 0≤ bi ≤ ∆ − 1 for all i.

(S3) bi ≥ bi−1and bi ≥ bi+1for all odd i . (S4) bi = bi+2 for all i .

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We now define T_∆as the infinite tree (except that T1is K2) whose vertex set consisting all

∆-sequences and a vertex (b0, b1, . . . , bm) is adjacent to another vertex (c0, c1, . . . , cn) if and only if|m − n| = 1 and bi = cifor 0≤ i ≤ min{m, n}. Now, we have

Theorem 6. Suppose T is a tree of maximum degree∆ ≥ 1 and j ≥ k are positive

integers satisfying j≥ ∆k. Then, T is λj,k-minimal if and only if T is a subtree of T∆.

Proof. (⇒) ByTheorem 4, T has aλj,k-labeling g such that any vertexv is labeled by

g(v) = a_vj+ b_vk where a_v ∈ {0, 1} and b_v ∈ {0, 1, . . . , ∆ − 1} satisfying conditions

(D1) and (D2) inTheorem 4. We may view T as a tree rooted at a major vertexv0. Without loss of generality, we may assume g(v0) = 0 by theProposition 1. For any vertexv in T , there is a unique path fromv0tov, denoted by P : v0, v1, . . . , vm. According to (D1) and (D2),(b_v₀, b_v₁, . . . , b_v_m) is a ∆-sequence. Then there is a one-to-one function from V (T ) to V(T_∆) mapping v to the corresponding ∆-sequence (b_v₀, b_v₁, . . . , b_v_m). It is clear that this mapping is edge-preserving. Hence T is a subtree of T_∆.

(⇐) According toProposition 1,λj,k(T ) ≥ j + (∆ − 1)k. Since T is a subtree of T∆, it suffices to give an L( j, k)-labeling of T_∆whose span is j+ (∆ − 1)k. Defin f as follows: for any vertex(b0, b1, . . . , bm) in T∆, let

f(b0, b1, . . . , bm) =

j+ bmk, if m is odd;

bmk, if m is even.

By conditions (S1) to (S4) for the definition of a∆-sequence, it is straightforward to check that f is an L( j, k)-labeling of T_∆.

Note that we may determine whether a tree T of maximum degree ∆ ≥ 1 is λp,q -minimal for all positive integers p≥ q or not by just checking whether it is λ_∆,1-minimal. This can be done by findingλ_∆,1(T ) using the algorithm described in[1]. We are expecting a simpler algorithm from just checking whether T is a subtree of T_∆.

Acknowledgements

We thank the referee for many constructive suggestions. This paper was supported in part by the National Science Council under grant NSC89-2115-M009-037.

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