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 yj 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 yj 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 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 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 yj B T:
For the case when T 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 yj over all vertices x and y; and third, the edge span of f, which is the maximum of j f x ÿ f yj over all edges xy. Given T and G, the T-chromatic number wT G is the minimum order of a T-coloring of G, the T-span spT G is the minimum span of a T-coloring of G, and the T-edge span espT G is the minimum edge span of a T-coloring of G.
Cozzens and Roberts [1] showed that the T-chromatic number wT 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 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 fv0; v1; . . . ; vnÿ1g and the edge set
E Cd
n 6
0UiUnÿ1fvivj: j 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 fv0; v1; . . . ; vnÿ1g and
edge set E Cn 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) spT KoG U espT G U spT G U spT Kw G.
(3) If T is k ÿ 1-initial, i.e., T f0; 1; . . . ; k ÿ 1g U S where S contains no multi-ple of k, then spT G spT Kw G k w G ÿ 1.
Theorem 2. (Liu [4]) For any odd cycle Cn and T f0; 1; . . . ; k ÿ 1g;
espT Cn n 1kn ÿ 1
.
Figure 1 shows an example of Cn with T f0; 1; 2g for which wT C7
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 f0; 1; 2; . . . ; k ÿ 1g. We note that Cd
n G Kn for d V n2
j k
and espT Kn spT Kn k n ÿ 1.
Therefore, throughout this article we consider Cd
n only for d U n2
j k
ÿ 1 and assume n 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 espT Cd
n (Theorem
4), both of them imply the exact value of esp Cd
n when r 0 (Theorem 5). We
then give a better upper bound when gcd n; d 1 1 (Theorem 6) and a better lower bound when r V 1 (Theorem 7), both of them imply the exact value when r 1 (Theorem 8).
Lemma 3. If n m d 1 r with m V 2 and 0 U r U d, then w Cd
n d 1 and
w Cd n mn
l m
d 1 l mmr . Proof. It is easy to see that o Cd
n 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 ni n ÿ im
, we have n Xmÿ1 i0 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 d r 1 m and so n ÿ im V d 1. Hence w Cd n U mn l m . r
Theorem 4. If n m d 1 r with m V 2 and 0 U r U d, then dk U espT Cd
n U spT Cnd dk mr
l m k.
Proof. The theorem follows from Theorem 1 and Lemma 3. r
Theorem 5. If n m d 1 with m V 2, then espT Cd
n spT Cnd dk.
Proof. The theorem follows from Theorem 4 as r 0. r Theorem 6. Suppose n m d 1 r with m V 2 and 0 U r U d. If gcd n; d 1 1, then espT Cd
n U dk rkm
.
Proof. Since gcd n; d 1 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 vja1; vja2; . . . ;
vjam (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
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 md r. Then j f vja ÿ f vjbj bk m ÿ akm Vbkmÿak m ÿ 1 m V k ÿ 1 1 m; Ubk m ÿ 1m ÿakm U md rkm 1 ÿm1; 8 > > > < > > > : or j f vja ÿ f vjbj V k; U dk rkm : 8 > < > : Therefore, f is a T-coloring of Cd n and espT Cnd U dk rkm . r
Theorem 7. If n m d 1 r with m V 2 and 1 U r U d, then espT Cd n V
dk mk
.
Proof. Suppose espT Cd
n U dk mk
ÿ 1. Let f be a T-coloring for which espT Cd
n maxfj f vi ÿ f vjj : vivj A E Cndg. Note that the m 1 vertices
vi d1; 0 U i U m, are pairwise non-adjacent except for v0vm d1A E Cnd. Let
ei; j f vi d1 ÿ f vj d1 for 0 U i U j U m. Then
k U je0; mj Xmÿ1 i0 ei; i1 U X mÿ1 i0 jei; i1j
and so there exists at least one i such that jei; i1j V mk
. In other words, the set U fi : jei; i1j 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 d1is not adjacent to v i1 d1. Sort the d 2 values
f vj; i d 1 U j U i 1 d 1; into b1U b2U U bd2. If fb1; bd2g 0
f f vi d1; f v i1 d1g, then
esp Cd n V bd2ÿ b1 Xd1 j1 bj1ÿ bj V dk mk ; a contradiction. Hence, fb1; bd2g f f vi d1; f v i1 d1g and
jei; i1j j f vi d1 ÿ f v i1 d1j
Xd1 j1 bj1ÿ bj V d 1k: Also, bd2ÿ b2U esp Cnd U dk mk ÿ 1; bd1ÿ b1U esp Cnd U dk mk ÿ 1;
bi1ÿ biV k for 2 U i U d and so; bd1ÿ b2V d ÿ 1k:
Then jei; i1j bd2ÿ b1U d 1k 2 mk ÿ 2. In conclusion, d 1k U jei; i1j U d 1k 2 mk ÿ 2 for all i A U: On the other hand, jei; i1j 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; i1in U1(or U2) are of the same sign.
For the case jU1j > jU2j, we have
espT Cd n V jeo; mj X mÿ1 i0 ei; i1 V X i A U1 jei; i1j ÿ X i A U2 jei; i1j ÿ X i B U jei; i1j V jU1j d 1k ÿ jU2j d 1k 2 mk ÿ 2 ÿ m ÿ jUj mk ÿ 1 jU1j ÿ jU2j d 1k jU1j ÿ jU2j ÿ m mk ÿ 1 V d 1k 1 ÿ m mk ÿ 1 > dk mk ÿ 1 since k > m mk ÿ 1 ; a contradiction.
For the case jU1j jU2j, say Ui fi1; i2; . . . ; iag for i 1; 2. Then k je0; mj Xmÿ1 i0 ei; i1 U Xa j1 e1j;1j1 e2j;2j1 X i B U jei; i1j UXa j1 d 1k 2 mk ÿ 2 ÿ d 1k X i B U k m ÿ 1 a 2 mk ÿ 2 m ÿ 2a mk ÿ 1 m mk ÿ 1 < k; a contradiction. r
Theorem 8. If n m d 1 1 with m V 2, then espT Cd
n dk mk
.
Proof. The theorem follows from Theorems 6 and 7 and the fact that
gcd n; d 1 1. r
Note that Theorem 2 is a special case to the above theorem when n is odd and d 1. For the case where n V 5 is odd and d n ÿ 32 , we have r 1; m 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 espT Cn n ÿ 2k2
.
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