Radio coloring of a graph G with diameter d is an assignment f of positive integers to the vertices of G such that |f(u)-f(v)|≥1+d-d(u,v), where u and v are any two distinct vertices of G and d(u,v) is the distance between u and v. The number max {f(u):u∈V(G)} is called the span of f. The minimum of spans over all radio colorings of G is called radio number of G, denoted by rn(G). An m-distant tree T is a tree in which there is a path P of maximum length such that every vertex in V(T)∖V(P) is at the most distance m from P. This path P is called a central path. For every tree T, there is an integer m such that T is a m-distant tree. In this paper, we determine the radio number of some m-distant trees for any positive integer m≥2, and as a consequence of it, we find the radio number of a class of 1-distant trees (or caterpillars).
1. Introduction
The channel assignment problem is the problem of assigning frequencies to transmitters in some optimal manner and with no interferences; see Hale [1]. Chartrand et al. [2] introduced radio k-colorings of graphs which is a variation of Hale’s channel assignment problem, in which one seeks to assign positive integers to the vertices of a graph G subject to certain constraints involving the distance between the vertices. For any simple connected graph G with diameter d and a positive integer k, 1≤k≤d, specifically, a radio k-coloring of G is an assignment f of positive integers to the vertices of G such that |f(u)-f(v)|≥1+k-d(u,v), where u and v are any two distinct vertices of G and d(u,v) is the distance between u and v. The maximum color (positive integer) assigned by f to some vertex of G is called the span of f, denoted by rck(f). The minimum of spans of all possible radio k-colorings of G is called the radio k-chromatic number of G, denoted by rck(G). A radio k-coloring with span rck(G) is called minimal radio k-coloring of G. Radio k-colorings have been studied by many authors; see [3–9].
Although the positive integer k can have value in-between 1 and d, the case k=d has become a special interest for many authors. Radio d-coloring is simply called radio coloring and radio d-chromatic number is radio number. Here we concentrate on radio number of trees. Kchikech et al. [4] have found the exact value of the radio k-chromatic number of stars K1,n as n(k-1)+2 and have also given an upper bound for radio k-chromatic number, rck(T), k≥2, of an arbitrary non-star-tree T on n vertices as (n-1)(k-1). Liu [5] has given a lower bound for the radio number rn(T) of an n-vertex tree with diameter d as (n-1)(d+1)+1-2w(T), where w(T) is the weight of T defined as w(T)=minu∈V(T)∑v∈V(T)d(u,v). She also has characterized the trees achieving this bound. In the same paper, Liu considered spiders denoted by Sl1,l2,l3,…,lm, which are trees having a vertex v of degree m≥3, and m number of paths of length l1,l2,…,lm whose one end vertex is v and other ends are pendant vertices. She has given a lower bound for the radio number of Sl1,l2,l3,…,lm as ∑i=1mli(l1+l2-li)+(l1-l2)/2(l1-l2)/2+1, where l1≥l2≥⋯≥lm, and has also characterized the spiders achieving this bound. Li et al. [10] have determined the radio number of complete m-ary trees (m≥3) with height k (≥2), denoted by Tk,m, as (mk+2+mk+1-2km2+(2k-3)m+1)/(m-1)2.
In this paper, we determine the radio number of some m-distant trees for any positive integer m≥2, and as a consequence of it, we find the radio number of a class of 1-distant trees (or caterpillars).
2. Radio Numbers of Some m-Distant Trees
Recall that an m-distant tree T is a tree in which there is a path P of maximum length such that every vertex in V(T)∖V(P) is at the most distance m from P. This path P is called a central path. Since we consider the path P as a path of maximum length, the end vertices of P are of degree one vertices in the m-distant tree; that is, if P: a1a2⋯ak is a central path of T, then degT(a1)=degT(ak)=1. For every tree T, there is an integer m such that T is a m-distant tree. Usually 1-distant trees are known as caterpillars.
Before we present the main result of the paper, we give a definition and a lemma below which will be used in the sequel. From the definition of a radio coloring f, one observes that for any two vertices u and v, the quantity |f(u)-f(v)|-{1+d-d(u,v)} is an excess for f to achieve the minimum span. In the definition, we give notation for these excesses corresponding to pair of vertices. In the lemma, we show that to get an optimal radio k-coloring, one has to minimize this sum of excesses.
Definition 1.
For any radio coloring f of a simple connected graph G on n vertices and an ordering x1,x2,…,xn of vertices of G with f(xi)≤f(xi+1), 1≤i≤n-1, we define ϵi (or ϵif to specify the coloring f) ={f(xi)-f(xi-1)}-{1+d-d(xi,xi-1)}, 2≤i≤n. It is clear from the definition of radio coloring that ϵi≥0, for all i. With respect to the ordering of vertices of G induced by f, we denote d(f)=∑i=2nd(xi,xi-1). In other words, every radio coloring f is associated with a unique positive integer d(f) called distance sum of f.
Example 2.
In this example, we explain Definition 1.
In Figure 1, a radio coloring f of a tree T is given. The labels x1,x2,x3,…,x12 are an ordering of vertices of T with f(xi)≤f(xi+1), 1≤i≤11. Here ϵ2={f(x2)-f(x1)}-{1+7-d(x1,x2)}=(5-1)-(1+7-4)=0, ϵ3=0, ϵ4=2, ϵ5=1, ϵ6=0, ϵ7=0, ϵ8=1, ϵ9=0, ϵ10=0, ϵ11=2, ϵ12=0, and d(f)=∑i=212d(xi,xi-1)=d(x2,x1)+d(x3,x2)+d(x4,x3)+d(x5,x4)+d(x6,x5)+d(x7,x6) + d(x8,x7)+d(x9,x8)+d(x10,x9)+d(x11,x10)+d(x12,x11)=4+7+1+5+2+2+2 + 3+3+3+4=36.
A radio coloring of a tree of order 12 and diameter 7.
The following lemma gives the span of a radio coloring of a graph of order n in terms of n, d, distance sum, and ϵ’s sum.
Lemma 3.
For any radio coloring f of G, the span of frcd(f)=f(xn)=(n-1)(1+d)-∑i=2nd(xi,xi-1)+∑i=2nϵif+1, where xi’s are arranged as in Definition 1.
Since f(x1)=1, we get f(xn)=(n-1)(1+d)-∑i=2nd(xi,xi-1)+∑i=2nϵif+1.
Lemma 3 says that to obtain a minimal radio coloring of a graph, one should maximize d(f) and minimize ∑i=2nϵif over all possible radio colorings of the graph. Since this fact is the basic concept to construct a minimal radio coloring, we express it as the theorem below.
Theorem 4.
If g is a radio coloring of G such that ∑i=2nϵig=0 and d(g)=max{d(f): f is a radio coloring of G}, then g is a minimal radio coloring of G.
Proof.
For any radio coloring f of G, Lemma 3 gives that rcd(f)=(n-1)(1+d)-d(f)+∑i=2nϵif. Then
(1)rcdG=min{rcd(f):fisaradiocoloringofG}=min∑i=2n(n-1)(1+d)-d(f)minnn+∑i=2nϵif:fisaradiocoloringofG=(n-1)(1+d)-maxfdf+minf∑i=2nϵif=n-11+d-dgsinceϵi,i=2,3,…,nisnonnegative,∑i=2nϵig=0=rcdg.
Now, we determine the radio number of an m-distant tree T with diam(T)=2p-1, p≥2, m≤p-p-1, and the degrees of the vertices on the central path satisfy certain conditions (given in the theorem below).
Theorem 5.
Let T be an m-distant tree of order n with a central path P: a1a2⋯a2p-1a2p, and satisfy
m≤p-p-1;
deg(ai)=2, for i=2,3,…,m, 2p-m+1,2p-m+2,…,2p;
deg(aj)=deg(aj+p-m)=sj+2, sj≥0, j=m+1,m+2,…,p;
for every al, l=m+1,m+2,…,2p-m, the number of vertices at distance i and lying on a branch incident on al, 2≤i≤m, is a constant say ti (ti≥0).
Then rn(T)=2(p-m)(p+m)∑i=2mti-∑i=2m2iti+2(p+m)-4∑i=m+1psi+2p(p-1)+3.
Proof.
The idea is to define a radio coloring f of T and show that f is minimal by Theorem 4. We first give an algorithm to order the vertices of T.
Algorithm 6
Step I. We make an ordering x1,x2,…,x2p of the vertices on the central path as ap, a2p, ap-1, a2p-1, ap-2, a2p-2, …, a1, ap+1, that is, x1=ap, x2=a2p,…,x2p=ap+1.
Step II. Let xl1,xl2,…,xl2p-2m be equal to the vertices ap, a2p-m, ap-1, a2p-(m+1),…,am+1, ap+1, respectively. Let B=(b1(i),b2(i),…,b2p-2m(i)) be an ordered tuple of vertices in T∖P such that bj(i) is at distance i from P and lies on a branch incident on xlj, i=2,3,…,m, and j=1,2,…,2p-2m. So for any i, there are ti disjoint such tuples, i=2,3,…,m. Consider the sequence S: x2p+1,x2p+2,…,xr, where r=2p+2(p-m)∑i=2mti.
Step III. We take i=m. In this case, there are tm disjoint tuples B. Select an arbitrary such tuple B and use the first 2p-2m terms of the sequence S to name the vertices in B in order. For the next tuple B we use the next 2p-2m terms of the sequence S to name the vertices in B in order. We proceed like this until we cover all the tm disjoint tuples.
Step IV. We name the vertices in T∖P which are at distance m-1 from P, in the similar manner. We proceed like this until we name all the vertices in T∖P and are of distance 2 from P.
Step V. Consider the sequence S1:xr+1,xr+2,…,xn. The terms in the beginning of S1 are assigned (or named) to the distance one vertices in T∖P adjacent to xl1 and xl2 alternately (the number of distance one vertices in T∖P adjacent to xli and xli+1, 1≤i≤2p-2m-1 are equal). The next terms of S1 are assigned to the distance one vertices in T∖P adjacent to xl3 and xl4 alternately, and so on, till we name all the vertices of T. Observe that n=2p+(2p-2m)∑i=2mti+2∑i=m+1psi.
Now x1,x2,…,xn is an ordering of all the vertices of T.
Let f be a coloring to the vertices of T defined by
(2)f(x1)=1f(xi)=f(xi-1)+2p-d(xi,xi-1),2≤i≤n.
Before we prove that f is a minimal radio coloring of T, we give an illustration of f.
Example 7. In this example, we illustrate the above coloring f by considering the 3-distant tree T given in Figure 2
Here p=6, the central path P: a1a2⋯a12, m=3, t2=1, t3=2, s4=3, s5=1, s6=2, x1=a6, x2=a12, x3=a5, x4=a11, x5=a4, x6=a10, x7=a3, x8=a9, x9=a2, x10=a8, x11=a1, x12=a7, xl1=a6, xl2=a9, xl3=a5, xl4=a8, xl5=a4, xl6=a7, S: x13,x14,…,x30, and S1: x31,x32,…,x42. So the ordering of the vertices of this 3-distant tree is in Figure 3.
The assignment f is given in Figure 4.
Continuation of the Proof of Theorem. We first show that f is a radio coloring. We need to check that |f(xi)-f(xj)|≥1+(2p-1)-d(xi,xj), 1≤i≠j≤n (we call this as radio condition). From the definition of f, radio condition holds true for pair of vertices xi and xi-1, 2≤i≤n. For 1≤i≤2p,
(3)fxi+2-fxi=f(xi+2)-f(xi+1)+f(xi+1)-f(xi)=2p-d(xi+2,xi+1)+2p-d(xi+1,xi)=4p-{d(xi+2,xi+1)+d(xi+1,xi)}=4p-{2p+1}=2p-1≥1+2p-d(xi,xi+2).
Since
(4)fx2p+1-fx2p-1=f(x2p+1)-f(x2p)+f(x2p)-f(x2p-1)=2p-(m+1)+2p-(p)=3p-(m+1)≥2p-1,
radio condition holds true for all the pair of vertices xi and xj, where one of them is on the central path P and the other in T∖P.
Now,
(5)fx2p+3-fx2p+1=f(x2p+3)-f(x2p+2)+f(x2p+2)-f(x2p+1)=2p-d(x2p+2,x2p+3)+2p-d(x2p+1,x2p+2)=2p-{p-m+1+2m}+2p-{p-m+2m}=4p-(p-m+2m)4p-v+(p-m+1+2m)=2p-(2m+1)=2p-d(x2p+1,x2p+3).
Since x2p+4,x2p+5,…,x2p+2p-2m are at least 2m+1 distance apart from x2p+1 and have colors greater than f(x2p+3), radio condition automatically holds true, and
(6)fx2p+2p-m-fx2p+1=∑i=22(p-m)f(x2p+i)-f(x2p+i-1)=2p(2(p-m)-1)-∑i=22(p-m)d(x2p+i,x2p+i-1)=4p2-4pm-2p-2(p-m)-22(p+m+p+m+1)+p+m=2p2-(4m+2)p+2m(m+1)+1.
If x2p+2(p-m)+1 is at a distance m from P, then
(7)fx2p+2p-m+1-f(x2p+1)=fx2p+2p-m+1-fx2p+2p-m+2p2-4m+2p+2mm+1+1=2p-2m+1+2p2-4m+2p+2mm+1+1=2p-m2≥2p-p-p-12=2p-2=2p-dx2p+1,x2p+2p-m+1.
Since f(x2p+2(p-m)+1)-f(x2p+1)≥2p-2, radio condition holds true between x2p+1 and all other vertices.
Let i=2p+(2p-2m)tm. Then xi is the last vertex at distance m from P. Since, from the definition of f, radio condition holds true for xi and xi+1, and d(xi,xi+1)=m+1+m-1=2m≤d(xi,xi+j), 2≤j≤2p-2m-1, radio condition automatically holds true between xi and xi+j, 2≤j≤2p-2m-1.
Since f(xi+1)-f(xi)=2p-{m+(m-1)+1}=2(p-m), f(xi+2)-f(xi+1)=2p-{2(m-1)+p-m}=p-m+2, and f(xi+3)-f(xi+2)=2p-{2(m-1)+p-m+1}=p-m+1, we have
(8)fxi+2p-2m-fxi=∑l=22p-2mfxi+l-fxi+l-1+f(xi+1)-f(xi)=2p-2m2(p-m+2)+2p-2m-22(p-m+1)+2(p-m)=2p-m2+4(p-m)-1≥2p-1=2p-d(xi,xi+2p-2m).
In a similar manner, one can check the radio condition for the remaining pair of vertices. Therefore, f is a radio coloring. Next, we show that f is minimal. Let x and y be any two vertices of T. Then we get vertices ai and aj on the central path P such that x=ai or x is on a branch incident on ai, and y=aj or y is on a branch incident on aj. Then
(9)d(x,y)=|j-i|,ifx=aiandy=aj|j-i|+j1,ifx=aiandyliesonabranchandisj1distancefromaj|j-i|+j2,ify=ajandxliesonabranchandisj2distancefromai|j-i|+j1+j2,ifxandylieonbranchesandarej2andj1distancesfromaiandajrespectively.
So, each term in the distance sum of a radio coloring contains two indices from {1,2,…,2p} with different signs because from (9), distance between every pair of vertices contains |j-i|, i,j∈{1,2,…,2p}. Since there are n-1 terms in the distance sum, it contains 2(n-1) elements from the set {1,2,…,2p} with half positive and half negative sign. Also, the indices 1,2,3,…,m and 2p-m-1,2p-m-2,…,2p-1,2p occur at most twice, and the index j∈{m+1,m+2,…,2p-m} occur at most 2∑i=2mti+sj+2 times, where deg(aj)=deg(aj+p-m)=sj+2. So, the possible maximum distance sum is 2(∑i=2mti+sm+2+1)p+2-m+2 + 2∑i=2mti+sm+3+1p+3-m+3+⋯+2∑i=2mti+sp-1+1[(2p-(m+1))-(p-1)] + 2∑i=2mti+sp+1(2p-m)-2∑i=2mti+sp+(1/2)(p) + 2∑i=2mti+sm+1+(1/2)(p+1)-2∑i=2mti+sm+1+1(m+1)+2(2p+2p-1+2p-2 + ⋯+2p-(m-1))-2(1+2+3+⋯+m)+2(p-m)∑i=2m2iti+2∑i=m+1p2si-1.
Now, the distance sum
(10)df=∑i=2nd(xi,xi-1)=∑i=22pdxi,xi-1+∑i=2p+12p+(2p-2m)tmdxi,xi-1+∑i=2p+(2p-2m)tm+12p+(2p-2m)tm+(2p-2m)tm-1dxi,xi-1+⋯+∑i=2p+(2p-2m)tm+(2p-2m)tm-1+⋯+(2p-2m)t3+12p+(2p-2m)tm+(2p-2m)tm-1+⋯+(2p-2m)t3+(2p-2m)t2dxi,xi-1+∑i=2p+(2p-2m)tm+(2p-2m)tm-1+⋯+(2p-2m)t3+(2p-2m)t2+1ndxi,xi-1,
where n=2p+∑i=2m(2p-2m)ti+2∑i=m+1psi coincides with the possible maximum distance sum above and is equal to 2(p-m)(p-m)∑i=2mti+∑i=2m2iti+2(p-m)+4∑i=m+1p2si+2p2-2.
One observes that ∑i=2nϵif=0. Therefore, from Theorem 4, the radio coloring f is minimal and
(11)rc2p-1T=rc2p-1(f)=n-11+2p-1-∑i=2ndxi,xi-1+1=2p+2p-m∑i=2mti+2∑i=m+1psi-12p-∑i=2ndxi,xi-1+1=2p-mp+m∑i=2mti-∑i=2m2iti+2(p+m)-4∑i=m+1psi+2p(p-1)+3.
Tree T to illustrate Theorem 5.
Ordering of vertices of T obtained from Theorem 5.
Radio coloring of T obtained from Theorem 5.
As a consequence of the above theorem, we determine the radio number of a class of caterpillars (1-distant trees). In the corollary below, we find radio number of caterpillars of odd diameter in which the degrees of every pair of nonpendant vertices on the central path lying at distance p-1 apart have the same degree (where 2p is the total number of vertices on the central path).
Corollary 8.
Let C be a caterpillar of order n and with a central path P:a1a2a3⋯a2p. If deg(ai)=deg(ap+i-1)=si+2, si≥0, i=2,3,…,p, then
(12)rn(C)=2p-1∑i=2psi+pp-1+3.
Proof.
This is m=1 case of Theorem 5. The ordering of vertices in this case includes Step I and Step V of algorithm in the proof of Theorem 5 with only variation that if a vertex on the central path is not adjacent to any pendant vertex, then we move to the next possible vertex.
Example 9.
In this example, we illustrate Corollary 8 by considering the caterpillar given in Figure 5.
A caterpillar to illustrate Corollary 8.
Here p=6, the central path P: a1a2⋯a12 and s2=2,s3=0,s4=3,s5=2. So the ordering of vertices of the caterpillar is illustrated in Figure 6 and the coloring f is given in Figure 7.
Ordering of vertices of the caterpillar given in Figure 5.
Radio coloring f of the caterpillar given in Figure 5.
The corollary below is also a consequence of Theorem 5 in which we find the radio number of caterpillars of odd diameter in which all nonpendant vertices on the central path are of the same degree.
Corollary 10.
Let C be a caterpillar of order n with a central path P: a1a2a3⋯a2p. If deg(ai)=s+2, i=2,3,…,2p-1, s≥0, then rn(C)=2[s(p-1)2+p(p-1)]+3.
Proof.
One can prove this result by substituting si=s, i=2,3,…,p in Corollary 8.
Example 11.
In this example we illustrate Corollary 10 by considering the caterpillar given in Figure 8.
A caterpillar for illustration of Corollary 10.
Here p=6, the central path P: a1a2⋯a12 and t=2. So the ordering of vertices of the caterpillar is illustrated in Figure 9 and the coloring f is given in Figure 10.
Ordering of vertices of the caterpillar given in Figure 8.
Minimal radio coloring of the caterpillar given in Figure 8.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors are thankful to the referee for his/her valuable comments and suggestions which improved the presentation of the paper.
HaleW. K.Frequency assignment: theory and applications198068121497151410.1109/proc.1980.118992-s2.0-0019213982ChartrandG.ErwinD.ZhangP.HararyF.Radio labelings of graphs2001337785MR1913399ChartrandG.ErwinD.ZhangP.A graph labeling problem suggested by FM channel restrictions2005434357MR2116390KchikechM.KhennoufaR.TogniO.Linear and cyclic radio k-labelings of trees200727110512310.7151/dmgt.1348MR2321426LiuD. D.Radio number for trees200830871153116410.1016/j.disc.2007.03.066MR23823542-s2.0-38149082016LiuD. D.ZhuX.Multilevel distance labelings for paths and cycles20051936106212-s2.0-33747193339MR219128310.1137/S0895480102417768PanigrahiP.A survey on radio k-colorings of graphs200961161169MR2533244KolaS. R.PanigrahiP.Improved lower bound for radio k-chromatic number of hypercube Qn20106021312140ZhangP.Radio labelings of cycles2002652132MR19380872-s2.0-0142231044LiX.MakV.ZhouS.Optimal radio labellings of complete m-ary trees2010158550751510.1016/j.dam.2009.11.014