T-colorings and T-edge spans of graphs

Graphs and

Combinatorics

( Springer-Verlag 1999

T-Colorings and T-Edge Spans of Graphs*

Shin-Jie Hu, Su-Tzu Juan, and Gerard J. Chang

Department of Applied Mathematics, National Chiao Tung University, Hsinchu 30050, Taiwan. e-mail: gjchang@math.nctu.edu.tw

Abstract. Suppose G is a graph and T is a set of non-negative integers that contains 0. A T-coloring of G is an assignment of a non-negative integer f …x† to each vertex x of G such that j f …x† ÿ f …y†j B T whenever xy A E…G†. The edge span of a T-coloring f is the maximum value of j f …x† ÿ f …y†j over all edges xy, and the T-edge span of a graph G is the minimum value of the edge span of a T-coloring of G. This paper studies the T-edge span of the dth power Cd

n of the n-cycle Cn for T ˆ f0; 1; 2; . . . ; k ÿ 1g. In particular, we ®nd the

exact value of the T-edge span of Cd

n for n 1 0 or 1 (mod d ‡ 1), and lower and upper

bounds for other cases.

1. Introduction

T-colorings were introduced by Hale [3] in connection with the channel assignment problem in communications. In this problem, there are n transmitters x1; x2; . . . ; xn

situated in a region. We wish to assign to each transmitter x a frequency f …x†. Some of the transmitters interfere because of proximity, meteorological, or other reasons. To avoid interference, two interfering transmitters must be assigned fre-quencies such that the absolute di¨erence of their frefre-quencies does not belong to the forbidden set T of non-negative integers and T contains 0. The objective is to make a frequency assignment that is e½cient according to certain criteria, while satisfying the above constraint.

To formulate the channel assignment problem graph-theoretically, we con-struct a graph G in which V…G† ˆ fx1; x2; . . . ; xng, and there is an edge between

transmitters xiand xj if and only if they interfere. Given graph G and a set T of

non-negative integers and T contains 0, a T-coloring of G is a function f from V…G† to the set of non-negative integers such that

xy A E…G† implies j f …x† ÿ f …y†j B T:

For the case when T ˆ f0g, T-coloring is the ordinary vertex coloring.

In channel assignments, the objective is to allocate the channels e½ciently. From the T-coloring standpoint, three criteria are important for measuring the

e½ciency: ®rst, the order of a T-coloring, which is the number of di¨erent colors used in f; second, the span of f, which is the maximum of j f …x† ÿ f …y†j over all vertices x and y; and third, the edge span of f, which is the maximum of j f …x† ÿ f …y†j over all edges xy. Given T and G, the T-chromatic number w_T…G† is the minimum order of a T-coloring of G, the T-span sp_T…G† is the minimum span of a T-coloring of G, and the T-edge span esp_T…G† is the minimum edge span of a T-coloring of G.

Cozzens and Roberts [1] showed that the T-chromatic number w_T…G† is equal to the chromatic number w…G†, which is the minimum number of colors needed to color the vertices of G so that adjacent vertices have di¨erent colors. The param-eter T-span of graphs has been studied extensively; for a good survey, see [6]; for recent results, see [2, 5, 7]. However, comparing to T-spans, there are relatively fewer known results about T-edge spans of graphs, see [1, 4].

Cozzens and Roberts [1] raised the problem of computing T-edge spans of non-perfect graphs when T ˆ f0; 1; 2; . . . ; k ÿ 1g. Liu [4] studied this problem for odd cycles. In this article, we consider Cd

n, the dth power of the n-cycle Cn. The

graph Cd

n has the vertex set V…Cnd† ˆ fv0; v1; . . . ; vnÿ1g and the edge set

E…Cd

n† ˆ 6

0UiUnÿ1fvivj: j ˆ i ‡ 1; i ‡ 2; . . . ; i ‡ dg;

where the index j for vjis taken modulo n. We ®nd the exact value of espT…Cnd† for

n 1 0 or 1 (mod d ‡ 1), and lower and upper bounds for other cases. 2. Previous results

In this section, we quote some known results about T-spans and T-edge spans, some of which will be used in Section 3.

The clique number o…G† of G is the maximum order of a clique (complete graph), a set of pairwise adjacent vertices. A complete graph of order n is denoted by Kn. The n-cycle is the graph Cn with vertex set V…Cn† ˆ fv0; v1; . . . ; vnÿ1g and

edge set E…Cn† ˆ fv0v1; v1v2; . . . ; vnÿ2vnÿ1; vnÿ1v0g. Note that Cn1is Cn.

The following are some known results on T-spans and T-edge spans.

Theorem 1. (Cozzens and Roberts [1]) The following statements hold for all graphs G and sets T.

(1) w…G† ÿ 1 U espT…G† U spT…G†.

(2) sp_T…KoG†† U espT…G† U spT…G† U spT…Kw…G††.

(3) If T is …k ÿ 1†-initial, i.e., T ˆ f0; 1; . . . ; k ÿ 1g U S where S contains no multi-ple of k, then sp_T…G† ˆ sp_T…Kw…G†† ˆ k…w…G† ÿ 1†.

Theorem 2. (Liu [4]) For any odd cycle Cn and T ˆ f0; 1; . . . ; k ÿ 1g;

esp_T…Cn† ˆ …n ‡ 1†k_{n ÿ 1}

 

.

Figure 1 shows an example of Cn with T ˆ f0; 1; 2g for which wT…C7† ˆ

3. Edge spans for powers of n-cycles

This section gives results for T-edge spans of Cd

n for the …k ÿ 1†-initial set

T ˆ f0; 1; 2; . . . ; k ÿ 1g. We note that Cd

n G Kn for d V n₂

j k

and esp_T…Kn† ˆ spT…Kn† ˆ k…n ÿ 1†.

Therefore, throughout this article we consider Cd

n only for d U n₂

j k

ÿ 1 and assume n ˆ m…d ‡ 1† ‡ r, where m V 2 and 0 U r U d. Our main results are as follows. First, we give an upper bound and a lower bound for esp_T…Cd

n† (Theorem

4), both of them imply the exact value of esp…Cd

n† when r ˆ 0 (Theorem 5). We

then give a better upper bound when gcd…n; d ‡ 1† ˆ 1 (Theorem 6) and a better lower bound when r V 1 (Theorem 7), both of them imply the exact value when r ˆ 1 (Theorem 8).

Lemma 3. If n ˆ m…d ‡ 1† ‡ r with m V 2 and 0 U r U d, then w…Cd

n† ˆ d ‡ 1 and

w…Cd n† ˆ _mn

l m

ˆ d ‡ 1 ‡l m_mr . Proof. It is easy to see that o…Cd

n† ˆ d ‡ 1 since d ‡ 1 U n₂

j k

; and w…Cd n† V _mn

l m since any independent set of Cd

n contains at most m vertices. Letting niˆ n ÿ i_m

  , we have n ˆXmÿ1 iˆ0 ni:

Color the n vertices of Cd

n as 1; 2; . . . ; n0; 1; 2; . . . ; n1; 1; 2; . . . ; n2; . . . ; 1; 2; . . . ; nmÿ1.

This coloring is a proper vertex coloring since eachn ÿ i m V n ÿ m ‡ 1 m ˆ d ‡ r ‡ 1 m and so n ÿ i_m   V d ‡ 1. Hence w…Cd n† U _mn l m . r

Theorem 4. If n ˆ m…d ‡ 1† ‡ r with m V 2 and 0 U r U d, then dk U esp_T…Cd

n† U spT…Cnd† ˆ dk ‡ _mr

l m k.

Proof. The theorem follows from Theorem 1 and Lemma 3. r

Theorem 5. If n ˆ m…d ‡ 1† with m V 2, then esp_T…Cd

n† ˆ spT…Cnd† ˆ dk.

Proof. The theorem follows from Theorem 4 as r ˆ 0. r Theorem 6. Suppose n ˆ m…d ‡ 1† ‡ r with m V 2 and 0 U r U d. If gcd…n; d ‡ 1† ˆ 1, then esp_T…Cd

n† U dk ‡ rk_m

  .

Proof. Since gcd…n; d ‡ 1† ˆ 1; d ‡ 1 is a generator of Zn using modulo n

addition, i.e., ji1 i …d ‡ 1† (mod n) for 0 U i U n ÿ 1 generates each integer

in f0; 1; . . . ; n ÿ 1g exactly once. In other words, we can consider V…Cd n† as

fvj0; vj1; . . . ; vjnÿ1g. Note that any m circularly consecutive vertices vja‡1; vja‡2; . . . ;

vja‡m (with indices a ‡ p considered modulo n) form an independent set in

Cd

n. Consequently, vjavjb is not an edge when 0 U a < b U n ÿ 1 with 1 U

minfb ÿ a; n ‡ a ÿ bg U m ÿ 1.

Now, consider the function f on V…Cd

n† de®ned by f …vji† ˆ

ik m  

for 0 U i U n ÿ 1. We claim that f is a T-coloring. For any edge vjavjb with 0 U a <

b U n ÿ 1, according to the preceding discussion, minfb ÿ a; n ‡ a ÿ bg V m, i.e., m U b ÿ a U n ÿ m ˆ md ‡ r. Then j f …vja† ÿ f …vjb†j ˆ bk m   ÿ ak_m   Vbk_mÿak ‡ m ÿ 1 m V k ÿ 1 ‡ 1 m; Ubk ‡ m ÿ 1_m ÿak_m U…md ‡ r†k_m ‡ 1 ÿ_m1; 8 > > > < > > > : or j f …vja† ÿ f …vjb†j V k; U dk ‡ rk_m   : 8 > < > : Therefore, f is a T-coloring of Cd n and espT…Cnd† U dk ‡ rk_m   . r

Theorem 7. If n ˆ m…d ‡ 1† ‡ r with m V 2 and 1 U r U d, then esp_T…Cd n† V

dk ‡ _mk  

.

Proof. Suppose esp_T…Cd

n† U dk ‡ _mk

 

ÿ 1. Let f be a T-coloring for which esp_T…Cd

n† ˆ maxfj f …vi† ÿ f …vj†j : vivj A E…Cnd†g. Note that the m ‡ 1 vertices

vi…d‡1†; 0 U i U m, are pairwise non-adjacent except for v0vm…d‡1†A E…Cnd†. Let

ei; jˆ f …vi…d‡1†† ÿ f …vj…d‡1†† for 0 U i U j U m. Then

k U je0; mj ˆ Xmÿ1 iˆ0 ei; i‡1       U X mÿ1 iˆ0 jei; i‡1j

and so there exists at least one i such that jei; i‡1j V _mk

 

. In other words, the set U ˆ fi : jei; i‡1j V _mk

 

For any i A U, the d ‡ 2 vertices vj; i…d ‡ 1† U j U …i ‡ 1†…d ‡ 1†; are

pair-wise adjacent except that vi…d‡1†is not adjacent to v…i‡1†…d‡1†. Sort the d ‡ 2 values

f …vj†; i…d ‡ 1† U j U …i ‡ 1†…d ‡ 1†; into b1U b2U


U bd‡2. If fb1; bd‡2g 0

f f …vi…d‡1††; f …v…i‡1†…d‡1††g, then

esp…Cd n† V bd‡2ÿ b1ˆ Xd‡1 jˆ1 …bj‡1ÿ bj† V dk ‡ _mk   ; a contradiction. Hence, fb1; bd‡2g ˆ f f …vi…d‡1††; f …v…i‡1†…d‡1††g and

jei; i‡1j ˆ j f …vi…d‡1†† ÿ f …v…i‡1†…d‡1††j ˆ

Xd‡1 jˆ1 …bj‡1ÿ bj† V …d ‡ 1†k: Also, bd‡2ÿ b2U esp…Cnd† U dk ‡ _mk   ÿ 1; bd‡1ÿ b1U esp…Cnd† U dk ‡ _mk   ÿ 1;

bi‡1ÿ biV k for 2 U i U d and so; bd‡1ÿ b2V …d ÿ 1†k:

Then jei; i‡1j ˆ bd‡2ÿ b1U …d ‡ 1†k ‡ 2 _mk   ÿ 2. In conclusion, …d ‡ 1†k U jei; i‡1j U …d ‡ 1†k ‡ 2 _mk   ÿ 2 for all i A U: On the other hand, jei; i‡1j U _mk

 

ÿ 1 for all i B U. Let U be the disjoint union of U1and U2such that jU1j V jU2j and all ei; i‡1in U1(or U2) are of the same sign.

For the case jU1j > jU2j, we have

esp_T…Cd n† V jeo; mj ˆ X mÿ1 iˆ0 ei; i‡1       V X i A U1 jei; i‡1j ÿ X i A U2 jei; i‡1j ÿ X i B U jei; i‡1j V jU1j…d ‡ 1†k ÿ jU2j …d ‡ 1†k ‡ 2 _mk   ÿ 2   ÿ …m ÿ jUj† _mk   ÿ 1   ˆ …jU1j ÿ jU2j†…d ‡ 1†k ‡ …jU1j ÿ jU2j ÿ m† _mk   ÿ 1   V …d ‡ 1†k ‡ …1 ÿ m† _mk   ÿ 1   > dk ‡ _mk   ÿ 1 since k > m _mk   ÿ 1     ; a contradiction.

For the case jU1j ˆ jU2j, say Uiˆ fi1; i2; . . . ; iag for i ˆ 1; 2. Then k  je0; mj ˆ Xmÿ1 iˆ0 ei; i‡1       U Xa jˆ1 …e1j;1j‡1‡ e2j;2j‡1†       ‡ X i B U jei; i‡1j UXa jˆ1 …d ‡ 1†k ‡ 2 _mk   ÿ 2 ÿ …d ‡ 1†k   ‡X i B U k m   ÿ 1   ˆ a 2 _mk   ÿ 2   ‡ …m ÿ 2a† _mk   ÿ 1   ˆ m _mk   ÿ 1   < k; a contradiction. r

Theorem 8. If n ˆ m…d ‡ 1† ‡ 1 with m V 2, then esp_T…Cd

n† ˆ dk ‡ _mk

  .

Proof. The theorem follows from Theorems 6 and 7 and the fact that

gcd…n; d ‡ 1† ˆ 1. r

Note that Theorem 2 is a special case to the above theorem when n is odd and d ˆ 1. For the case where n V 5 is odd and d ˆn ÿ 3₂ , we have r ˆ 1; m ˆ 2, and Cd

n is isomorphic to the complement Cnof Cn. Thus, we have the following result.

Corollary 9. If n V 5 is odd, then esp_T…Cn† ˆ …n ÿ 2†k₂

 

.

Acknowledgments. The authors thank a referee and Daphne Der-Fen Liu for many useful suggestions on revising the paper.

References

1. Cozzens, N.B., and Roberts, F.S.: T-colorings of graphs and the channel assignment problem, Congr. Numerantium 35, 191±208 (1982)

2. Griggs, J.R., and Liu, D.D.-F.: Channel assignment problem for mutually adjacent sites, J. Comb. Theory, Ser. A 68, 169±183 (1994)

3. Hale, W.K.: Frequency assignment: theory and applications, Proc. IEEE 68, 1497± 1514 (1980)

4. Liu, D.D.-F.: Graph Homomorphisms and the Channel Assignment Problem, Ph.D. Thesis, Department of Mathematics, University of South Carolina, Columbia, SC, 1991

5. Liu, D.D.-F.: T-graphs and the channel assignment problem, Discrete Math. 161, 198± 205 (1996)

6. Roberts, F.S.: T-colorings of graphs: recent results and open problems, Discrete Math. 93, 229±245 (1991)

7. Tesman, B.A.: List T-colorings of graphs, Discrete Appl. Math. 45, 277±289 (1993)

Received: May 13, 1996 Revised: December 8, 1997