Radio number of the cartesian product of a tree and a complete graph

Payal Vasoya1, Devsi Bantva2
1Gujarat Technogical University, Ahmedabad – 382 424, Gujarat, India
2Lukhdhirji Engineering College, Morvi 363 642, Gujarat, India

Abstract

A radio labeling of a graph G is a mapping f:V(G){0,1,2,} such that |f(u)f(v)|diam(G)+1d(u,v) for every pair of distinct vertices u,v of G, where diam(G) is the diameter of G and d(u,v) is the distance between u and v in G. The radio number rn(G) of G is the smallest integer k such that G admits a radio labeling f with max{f(v):vV(G)}=k. In this paper, we give a lower bound for the radio number of the Cartesian product of a tree and a complete graph and give two necessary and sufficient conditions for the sharpness of the lower bound. We also give three sufficient conditions for the sharpness of the lower bound. We determine the radio number of the Cartesian product of a level-wise regular tree and a complete graph which attains the lower bound. The radio number of the Cartesian product of a path and a complete graph derived in [B. M. Kim, W. Hwang, and B. C. Song, Radio number for the product of a path and a complete graph, J. Comb. Optim., 30 (2015), 139–149] can be obtained using our results in a short way.

Keywords: radio labelling, radio number, cartesian product, tree, complete graph

1. Introduction

The channel assignment problem is the problem of assigning a channel to each transmitter in a radio network such that a set of constraints is satisfied and the span is minimized. The constraints for assigning channels to transmitters are usually determined by the geographic location of the transmitters; the closer the location, the stronger the interference might occur. In order to avoid stronger interference, the larger frequency gap between two assigned frequencies must be required. In [10], Hale designed the optimal labelling problem for graphs to deal with this channel assignment problem. In this model, the transmitters are represented by the vertices of a graph, and two vertices are adjacent if the corresponding transmitters are close to each other. Initially, only two levels of interference, namely avoidable and unavoidable, were considered which inspired Griggs and Yeh [8] to introduce the following concept: An L(2,1)-labelling of a graph G is a function f:V(G){0,1,2,} such that |f(u)f(v)|2 if d(u,v)=1 and |f(u)f(v)|1 if d(u,v)=2. The span of f is defined as max{|f(u)f(v)|:u,vV(G)}, and the λ-number (or the λ2,1-number) of G is the minimum span of an L(2,1)-labelling of G. The L(2,1)-labelling problem has been studied extensively in the past more than two decades, as one can find in the survey articles [5,19].

Denote the diameter of a graph G by diam(G) (the diameter of a graph G is diam(G)=max{d(u,v):u,vV(G)}). In [7,6], Chartrand et al. introduced the concept of the radio labelling problem by extending the condition on distance in L(2,1)-labelling from two to the maximum possible distance in a graph, namely the diameter of a graph.

Definition 1.1. A radio labelling of a graph G is a mapping f:V(G){0,1,2,} such that the following hold for every pair of distinct vertices u,v of G, (1)|f(u)f(v)|diam(G)+1d(u,v). The integer f(u) is called the label of u under f, and the span of f is defined as span(f)=max{|f(u)f(v)|:u,vV(G)}. The radio number of G is defined as rn(G)=minfspan(f) with minimum taken over all radio labellings f of G. A radio labelling f of G is called optimal if span(f)=rn(G).

Observe that the radio labelling problem is a min-max type optimization problem. Without loss of generality, we may always assume that any radio labelling assigns 0 to some vertex so that the span of a radio labelling is equal to the maximum label used. Since d(u,v)diam(G), any radio labelling always assigns different labels to distinct vertices. Therefore, a radio labelling f of graph G with order m induces the linear order (2)Vf:a0,a1,,am1, of the vertices of G such that (3)0=f(a0)<f(a1)<<f(am1)=span(f). A linear order a0,a1,,am1 of V(T) is called an optimal linear order if it induced by some optimal radio labelling f of T.

Determining the radio number of graphs is a tough and challenging task. The radio number is known only for very few graphs and much attention has been paid to special families of graphs. It turns out that even for some special graph families the problem may be difficult. For example, the radio number of paths was determined by Liu and Zhu in [17], and even for this basic graph family the problem is nontrivial. In [15,16], Liu and Xie determine the radio number for the square graph of paths and cycles. In [4], Benson et al. determined the radio number of all graphs of order n2 and diameter n2. In [13], Liu gave a lower bound for the radio number of trees and a necessary and sufficient condition for this bound to be achieved. The author also presented a special class of trees, namely spiders, achieving this lower bound. In [12], Li et al. determined the radio number of complete m-ary trees of height k, for k1,m2. In [9], Halász and Tuza gave a lower bound for the radio number of level-wise regular trees and proved further that this bound is tight when all the internal vertices have degree more than three. In [3], Bantva et al. gave a necessary and sufficient condition for the lower bound given in [13] to be tight along with two sufficient conditions for achieving this lower bound. Using these results, they also determined the radio number of three families of trees in [3]. In [1], Bantva determined the radio number of some trees obtained by applying a graph operation on given trees. In [14], Liu et al. improved the lower bound for the radio number of trees. Recently, in [2], Bantva and Liu gave a lower bound for the radio number of block graphs and, three necessary and sufficient conditions for achieving the lower bound. The authors gave three other sufficient conditions for achieving the lower bound and also discuss the radio number of line graphs of trees and block graphs in [2]. Using these results, the authors determine the radio number of extended star of blocks and level-wise regular block graphs in [2].

In this paper, we give a lower bound for the radio number of the Cartesian product of a path and a complete graph (Theorem 3.1). We also give two necessary and sufficient conditions (Theorems 3.2 and 3.3) and three other sufficient conditions (Theorem 3.4) for achieving the lower bound. Using these results, we determine the radio number of the Cartesian product of a level-wise regular tree and a complete graph. The radio number of the Cartesian product of a path and a complete graph given in [11] can be obtained using our results in a short way.

2. Preliminaries

We follow [17] for standard graph-theoretic terms and notations. In a graph G, the neighbourhood of any vV(G) is NG(v)={u:u is adjacent to v}. The distance between two vertices u and v in G, denoted by dG(u,v), is the length of the shortest path joining u and v in G. The diameter of a graph G, denoted by diam(G), is diam(G)=max{dG(u,v):u,vV(G)}. We drop the suffix in above-defined terms when G is clear in the context. A complete graph Kn is a graph on n vertices in which any two vertices are adjacent. A tree is a connected acyclic graph. We fix V(T)={u0,u1,,um1} for tree T of order m and V(Kn)={v0,v1,,vn1} throughout the paper. A vertex vV(T) is an internal vertex of T if it has degree greater than one and is a leaf otherwise. In [3,13], the weight of T from vV(T) is defined as wT(v)=uV(T)d(u,v), and the weight of T is defined as w(T)=min{wT(v):vV(T)}. A vertex vV(T) is a weight center [3,13] of T if wT(v)=w(T). Denote by W(T) the set of weight centers of T. In [13], the following is proved about W(T).

Lemma 2.1 . [13]] If r is a weight center of a tree T. Then each component of Tr contains at most |V(T)|/2 vertices.
Lemma 2.2. [13]] Every tree T has one or two weight centers, and T has two weight centers, say, W(T)={r,r} if and only if rr is an edge of T and Trr consists of two equal sized components.

A vertex u is called [3,13] an ancestor of a vertex v, or v is a descendent of u, if u is on the unique path joining a weight center and v. Let uV(T)W(T) be a vertex adjacent to a weight center x. The subtree of T induced by u and all its descendants is called the branch of T at u. Two vertices u,v of T are said to be in different branches if the path between them contains only one weight center and in opposite branches if the path joining them contains two weight centers. We view a tree T rooted at its weight center W(T): if W(T)={r}, then T is rooted at r; if W(T)={r,r}, then T is rooted at r and r in the sense that both r and r are at level 0. In either case, for each uV(T), define L(u)=min{d(u,x):xW(T)}, to indicate the level of u in T, and define L(T)=uV(T)L(u), the total level of T. For any u,vV(T), define ϕ(u,v)=max{L(x):x is a common ancestor of u and v}, δ(u,v)={1,if |W(T)|=2 and the (u,v)-path in T containsboth weight centers,0,otherwise.

Lemma 2.3 ([3, Lemma 2.1]). Let T be a tree with diameter d2. Then for any u,vV(T) the following hold:

(a) ϕ(u,v)0;

(b) ϕ(u,v)=0 if and only if u and v are in different or opposite branches;

(c) δ(u,v)=1 if and only if T has two weight centers and u and v are in opposite branches;

(d) the distance d(u,v) in T between u and v can be expressed as

(4)d(u,v)=L(u)+L(v)2ϕ(u,v)+δ(u,v).

Define ε(T)={1,if |W(T)|=1,0,if |W(T)|=2.

Let G=(V(G),E(G)) and H=(V(H),E(H)) be two graphs. The Cartesian product of G and H is the graph G◻H with V(G◻H)=V(G)×V(H) such that two vertices (a,b) and (c,d) are adjacent if a=c and (b,d)E(H) or b=d and (a,c)E(G). It is clear from the definition that dG◻H((x1,y1),(x2,y2))=dG(x1,x2)+dH(y1,y2).

Observation 2.4. For a tree T of order m(m2) and a complete graph Kn,

(a) |V(T◻Kn)|=mn, and

(b) diam(T◻Kn)=diam(T)+1.

Let V=(w0,w1,,wmn1) be an ordering of V(T◻Kn), where |T|=m. Then each wt(0tmn1) is an ordered pair (uit,vjt) with uitV(T) and vjtV(Kn). Hence each vertex ui,i=0,1,,m1 appears n times and each vertex vj,j=0,1,,n1 appears m times in T◻Kn. For wa=(uia,vja),wb=(uib,vjb)V(T◻Kn)(0a,bmn1), vja and vjb are called distinct if jajb. For any wa=(uia,vja),wb=(uib,vjb)V(T◻Kn)(0a,bmn1), the distance between wa and wb is given by (5)d(wa,wb)=L(uia)+L(uib)+δ(uia,uib)2ϕ(uia,uib)+d(vja,vjb).

3. A tight lower bound for rn(T◻Kn)

In this section, we continue to use terms and notations defined in the previous section. We first give a lower bound for the radio number of the Cartesian product of a tree and a complete graph. Next we give two necessary and sufficient conditions and three other sufficient conditions for achieving the lower bound.

Theorem 3.1. Let T be a tree of order m and diameter d2. Denote ε=ε(T). Then (6)rn(T◻Kn)(mn1)(d+ε)2nL(T).

Proof. It is enough to prove that any radio labelling of T◻Kn has no span less than that of the right-hand side of (6). Suppose f is any radio labelling of T◻Kn which induces an ordering Vf=(w0,w1,,wmn1) such that 0=f(w0)<f(w1)<<f(wmn1). Denote diam(T◻Kn)=d. By the definition of a radio labelling, f(wt+1)f(wt)d+1d(wt,wt+1) for 0tmn2. Since d=d+1 by Observation 2.4, we have f(wt+1)f(wt)d+2d(wt,wt+1) for 0tmn2. Summing up these mn1 inequalities, we obtain (7)span(f)=f(wmn1)(mn1)(d+2)t=0mn2d(wt,wt+1).

Case 1. |W(T)|=1. In this case, δ(uit,uit+1)=0 by the definition of δ, ϕ(uit,uit+1)0 and 0d(vjt,vjt+1)1 for all 0tmn2. Hence, using (5), we obtain t=0mn2d(wt,wt+1)=t=0mn2[L(uit)+L(uit+1)2ϕ(uit,uit+1)+d(vjt,vjt+1)]t=0mn2[L(uit)+L(uit+1)+d(vjt,vjt+1)]2ni=0m1L(ui)+(mn1)L(ui0)L(uimn1)2nL(T)+mn1. Substituting this into (7), we have span(f)(mn1)(d+1)2nL(T).

Case 2. |W(T)|=2. In this case, 0δ(uit,uit+1)1, ϕ(uit,uit+1)0 and 0d(vjt,vjt+1)1 for all 0tmn2. Hence, using (5), we obtain t=0mn2d(wt,wt+1)=t=0mn2[L(uit)+L(uit+1)+δ(uit,uit+1)2ϕ(uit,uit+1)+d(vjt,vjt+1)]t=0mn2[L(uit)+L(uit+1)+1+d(vjt,vjt+1)]2ni=0m1L(ui)+(mn1)+(mn1)L(ui0)L(uimn1)2nL(T)+2(mn1). Substituting this into (7), we have span(f)(mn1)d2nL(T)◻

Theorem 3.2. Let T be a tree of order m and diameter d2. Denote ε=ε(T). Then (8)rn(T◻Kn)=(mn1)(d+ε)2nL(T), if and only if there exists an ordering w0,w1,,wmn1 of V(T◻Kn) such that the following hold:

(a) L(ui0)=L(uimn1)=0;

(b) vjt and vjt+1 are distinct for all 0tmn2;

(c) for wa=(uia,vja),wb=(uib,vjb)(0a<bmn1), the distance between any two vertices uia and uib satisfies

(9)d(uia,uib)t=ab1[L(uit)+L(uit+1)(d+ε)]+(d+1).

Moreover, under these conditions (a)-(c), the mapping f defined by (10)f(w0)=0, (11)f(wt+1)=f(wt)+d+εL(uit)L(uit+1),0tmn2, is an optimal radio labelling of T◻Kn.

Proof. Necessity. Suppose that (8) holds. Note that (8) holds if equality hold in (1) and everywhere in the proof of Theorem 3.1. Again observe that if the equality holds everywhere in the proof of Theorem 3.1 then an ordering w0,w1,,wmn1 of V(T◻Kn) induced by f satisfies the followings: (1) L(ui0)=L(uimn1)=0, (2) uit and uit+1 are in different branches when |W(T)|=1 and in opposite branches when |W(T)|=2 for all 0tmn2, (3) vjt and vjt+1 are distinct for all 0tmn2. Hence in this case the definition of radio labelling f can be written as f(w0)=0 and f(wt+1)=f(wt)+d+εL(uit)L(uit+1)1=f(wt)+d+εL(uit)L(uit+1) for 0tmn2, where d=diam(T◻Kn). Summing this last equality for wa to wb(0a<bmn1), we obtain f(wb)f(wa)=t=ab1[d+εL(uit)L(uit+1)].

Since f is a radio labelling of T◻Kn, we have f(wb)f(wa)d+1d(wa,wb)=d+2d(wa,wb). Also note that d(wa,wb)=d(uia,uib)+1 by (5). Substituting these into the above equation, we obtain d(uia,uib)t=ab1[L(uit)+L(uit+1)(d+ε)]+(d+1).

Sufficiency. Suppose there exists an ordering (w0, w1, , wmn1) of V(T◻Kn) satisfies the conditions (a)-(c) and f is defined by (10) and (11). Note that it is enough to prove that f is a radio labelling of T◻Kn and the span of f is the right-hand side of (8). Denote diam(T◻Kn)=d. Let wa and wb(0a<bmn1) be two arbitrary vertices. f(wb)f(wa)=t=ab1(f(wt+1)f(wt)),=t=ab1(d+εL(uit)L(uit+1)),=d+1d(uia,uib),=d+2(d(uia,uib)+1),=d+1d(wa,wb). The span of f is span(f)=f(wmn1)f(w0),=t=0mn2(f(wt+1)f(wt)),=t=0mn2(d+εL(uit)L(uit+1)),=(mn1)(d+ε)2t=0mn1L(uit)+L(ui0)+L(uimn1),=(mn1)(d+ε)2nL(T). ◻

Theorem 3.3. Let T be a tree of order m and diameter d2. Denote ε=ε(T). Then (12)rn(T◻Kn)=(mn1)(d+ε)2nL(T), if and only if there exists an ordering w0,w1,,wmn1 of V(T◻Kn) with wt=(uit,vjt),0tmn1 such that the following all hold:

(a) L(ui0)=L(uimn1)=0;

(b) vjt and vjt+1 are distinct for 0tmn2;

(c) uit and uit+1 are in different branches when |W(T)|=1 and in opposite branches when |W(T)|=2 for 0tmn2;

(d) L(uit)(d+1)/2 when |W(T)|=1 and L(uit)(d1)/2 when |W(T)|=2 for all 0tmn1;

(e) for any wa=(uia,vja),wb=(uib,vjb)(0a<bmn1) such that uia and uib are in the same branch of T, uia and uib satisfy (13)ϕ(uia,uib)(ba12)(d+ε)t=a+1b1L(uit)(1ε2).

Proof. Necessity. Suppose (12) holds. Then there exists an ordering (w0, w1, , wmn1) of T◻Kn such that the conditions (a)-(c) of Theorem 3.2 holds. Hence the conditions (a) and (b) are satisfied. Taking a=t and b=t+1 in (9), we obtain d(wt,wt+1)=d(uit,uit+1)+d(vjt,vjt+1)=L(uit)+L(uit+1)+1ε+1. Hence we have d(uit,uit+1)=L(uit)+L(uit+1)+1ε for all 0tmn2 as 0d(uit,uit+1)L(uit)+L(uit+1)+1ε and 0d(vjt,vjt+1)1. Therefore, by Lemma 2.3, uit and uit+1 are in different branches when |W(T)|=1 and in opposite branches when |W(T)|=2 for all 0tmn2. Hence the condition (c) is satisfied. Since L(ui0)=L(uimn1)=0, it is clear that L(uit)<(d+1)/2 for t=0,mn1. For 1tmn2, considering uit1 and uit+1 in (9) and using (5), we obtain 2L(uit)(d+ε)(1ε)+δ(uit1,uit+1)2ϕ(uit1,uit+1).

In above equation, observe that when |W(T)|=1 then δ(uit1,uit+1)=0 by the definition of δ and when |W(T)|=2 then uit1 and uit+1 are in different or in the same branch of T and hence δ(uit1,uit+1)=0. Thus we have 2L(uit)d+2ε1.

Hence, the condition (d) is satisfied. To prove (e), let wa=(uia,vja) and wb=(uib,vjb)(0a<bmn1) be such that uia and uib are in the same branch of T. Then δ(uia,uib)=0. Using (5) in (9), (13) can be obtained easily from (9).

Sufficiency. Suppose there exists an ordering (w0,w1,,wmn1) of V(T◻Kn) such that conditions (a)-(e) hold. To prove (12) holds, we show that an ordering w0,w1,,wmn1 satisfies the conditions (a)-(c) of Theorem 3.2. Since the conditions (a) and (b) are identical with the conditions (a) and (b) of Theorem 3.2, we need to prove the condition (9) only. Denote the right-hand side of (9) by Sa,b. Let wa=(uia,vja),wb=(uib,vjb)(0a<bmn1) be any two vertices. If uia and uib are in opposite branches, then d(uia,uib)=L(uia)+L(uib)+1. Hence, Sa,b=L(uia)+L(uib)+2t=a+1b1L(uit)(ba)d+d+1L(uia)+L(uib)+1+2(ba1)((d1)/2)(ba1)d=L(uia)+L(uib)+1(ba1)L(uia)+L(uib)+1=d(uia,uib). If uia and uib are in different branches, then d(uia,uib)=L(uia)+L(uib) and Sa,b=L(uia)+L(uib)+2t=a+1b1L(uit)(ba1)(d+ε)+(1ε). If |W(T)|=1, then note that L(uit)(d+1)/2 for all 0tmn1 and hence Sa,bL(uia)+L(uib)+2(ba1)((d+1)/2)(ba1)(d+1)=L(uia)+L(uib)=d(uia,uib). If |W(T)|=2, then note that ba11 and L(uit)(d1)/2 for all 0tmn1 and hence Sa,bL(uia)+L(uib)+2(ba1)((d1)/2)(ba1)d+1=L(uia)+L(uib)+1(baa)L(uia)+L(uib)=d(uia,uib). If uia and uib are in the same branch, then (9) can be obtained easily using (13). This completes the proof. ◻

Theorem 3.4. Let T be a tree of order m and diameter d2. Denote ε=ε(T). Then (14)rn(T◻Kn)=(mn1)(d+ε)2nL(T), if there exists an ordering w0,w1,,wmn1 of V(T◻Kn) such that

(a) L(ui0)=L(uimn1)=0,

(b) vjt and vjt+1 are distinct for all 0tmn2,

(c) uit and uit+1 are in different branches when |W(T)|=1 and in opposite branches when |W(T)|=2 for all 0tmn2,

and one of the following holds:

(d) min{d(uit,uit+1),d(uit+1,uit+2)}(d+1ε)/2 for all 0tmn3;

(e) d(uit,uit+1)(d+1+ε)/2 for all 0tmn2;

(f) for all 0tmn1, L(uit)(d+1)/2 when |W(T)|=1 and L(uit)(d1)/2 when |W(T)|=2 and, if wa=(uia,vja) and wb=(uib,vjb)(0a<bmn1) such that uia and uib are in the same branch of T then bad.

Proof. We show that if (a)-(c) and one of (d)-(f) holds for an ordering w0,w1,,wmn1 such that wt=(uit,vjt),0tmn1, then (a)-(e) of Theorem 3.3 are satisfied. Since the conditions (a)-(c) are identical in both Theorems 3.3 and 3.4, we need to verify the conditions (d) and (e) of Theorem 3.3 only. Denote the right-hand side of (13) by Pa,b. We consider the following two cases.

Case 1. |W(T)|=1. In this case, recall that ε=1 and δ(uit,uit+1)=0 for all 0tmn2 by the definition of δ.

Subcase 1.1. Suppose (a)-(d) hold. It is clear that L(ui0)=L(uimn1)=0(d+1)/2 and for 1tmn2, L(uit)min{d(uit,uit+1),d(uit+1,uit+2)}d/2<(d+1)/2. Let wa=(uia,vja) and wb=(uib,vjb),0a<bmn1 be two arbitrary vertices such that uia and uib are in the same branch of T. If ba4, then Pa,b3(d+1)/2d/2(d+1)/2=d/2+1ϕ(uia,uib). Assume ba=3. If L(uia+1)+L(uia+2)d/2, then Pa,b(d+1)d/2=d/2+1ϕ(uia,uib). If L(uia+1)+L(uia+2)>d/2, then L(uia+2)+L(uia+3)d/2 and hence L(uia+2)d/2L(uia+3). Therefore, Pa,bd+1L(uia+1)L(uia+2)d+1(d+1)/2d/2+L(uia+3)=L(uia+3)+1/2ϕ(uia,uib). Assume ba=2. Then either L(uia)+L(uia+1)d/2 or L(uia+1)+L(uia+2)d/2. Without loss of generality, we may assume that L(uia)+L(uia+1)d/2. Then L(uia+1)d/2L(uia). Hence, Pa,b=(d+1)/2L(uia+1)(d+1)/2d/2+L(uia)=L(uia)+(1/2)ϕ(uia,uib).

Subcase 1.2. Suppose (a)-(c) and (e) hold. Note that L(ui0)=L(uimn1)=0(d+1)/2. Since L(uit)1 and L(uit)+L(uit+1)=d(uit,uit+1)(d+2)/2 for 0tmn1, we obtain L(uit)(d+1)/2 for all 0tmn1.

Let wa=(uia,vja) and wb=(uib,vjb),0a<bmn1 be two arbitrary vertices such that uia and uib are in the same branch of T. If ba3, then Pa,bd+1L(uia+1)L(uia+2)d+1(d+2)/2=d/2ϕ(uia,uib). Assume ba=2. Note that L(uia)+L(uia+1)(d+2)/2 and hence L(uia+1)(d+2)/2L(uia). Therefore, Pa,b=(d+1)/2L(uia+1)(d+1)/2((d+2)/2L(uia))=L(uia)(1/2)ϕ(uia,uib).

Subcase 1.3. Suppose (a)-(c) and (f) hold. Assume d is even. Then L(uit)d/2 for all 0tmn1 and hence Pi,j((ba1)/2)(d+1)(ba1)(d/2)(ba1)/2(d1)/2ϕ(uia,uib). Assume d is odd. Then note that only for one vertex uiq, L(uiq)=(d+1)/2 from consecutive ba vertices and for all other vertices uit, L(uit)d/2. Hence, Pa,b((ba1)/2)(d+1)(ba2)(d/2)(d+1)/2(ba2)/2(d2)/2ϕ(uia,uib).

Case 2. |W(T)|=2. In this case, recall that ε=0 and δ(uit,uit+1)=1 for all 0tmn1 by the definition of the ordering w0,w1,,wmn1 of V(T◻Kn).

Subcase 2.1. Suppose (a)-(d) hold. It is clear that L(ui0)=L(uimn1)=0(d1)/2 and for 1tmn2, L(uit)min{d(uit1,uit),d(uit,uit+1)}1(d+1)/21=(d1)/2. Let wa=(uia,vja) and wb=(uib,vjb),0a<bmn1 be two arbitrary vertices such that uia and uib are in the same branch of T. If ba4, then Pa,b3d/2(d1)1/2=(d+1)/2ϕ(uia,uib). Assume ba=3. If L(uia+1)+L(uia+2)(d1)/2, then Pa,bd(d1)/21/2=d/2ϕ(uia,uib). If L(uia+1)+L(uia+2)>(d1)/2, then L(uia)+L(uia+1)(d1)/2 and hence L(uia+1)(d1)/2L(uia). Therefore, Pa,bd((d1)/2L(uia))(d1)/21/2=L(uia)+1/2ϕ(uia,uib). Assume ba=2. Then either L(uia)+L(uia+1)(d1)/2 or L(uia+1)+L(uia+2)(d1)/2. Without loss of generality, we may assume that L(uia+1)+L(uia+2)(d1)/2. Then L(uia+1)(d1)/2L(uia+2). Hence, Pa,b=d/2L(uia+1)1/2d/2((d1)/2L(ia+2))1/2=L(uia+2)ϕ(uia,uib).

Subcase 2.2. Suppose (a)-(c) and (e) hold. Note that L(ui0)=L(uimn1)=0(d1)/2. Since L(uit)1 and L(uit)+L(uit+1)=d(uit,uit+1)1(d+1)/21=(d1)/2 for 0tmn1, we obtain L(uit)(d1)/2 for all 0tmn1.

Let wa=(uia,vja) and wb=(uib,vjb),0a<bmn1 be two arbitrary vertices such that uia and uib are in the same branch of T. If ba3, then Pa,bdL(uia+1)L(uia+2)1/2d(d1)/21/2=d/2ϕ(uia,uib). Assume that ba=2. Then note that L(uia)+L(uia+1)(d1)/2 and hence L(uia)(d1)/2L(uia+1). Hence, Pa,bd/2L(uit)(1/2)d/2[(d1)/2L(uia+1)](1/2)=L(uia+1)ϕ(uia,uib).

Subcase 2.3. Suppose (a)-(c) and (f) hold. Assume d is even. Then L(uit)(d2)/2 for all 0tmn1 and hence Pa,b((ba1)/2)d(ba1)((d2)/2)1/2=ba(3/2)d(3/2)ϕ(uia,uib). Assume d is odd. Then note that only for one vertex uiq, L(uiq)(d1)/2 from ba consecutive vertices and for all other vertices uit, L(uit)(d3)/2. Hence, Pa,b((ba1)/2)d(ba2)((d3)/2)(d1)/2(1/2)=3(ba2)/23(d2)/2ϕ(uia,uib)◻

4. Radio number of the Cartesian product of a level-wise regular tree and a complete graph

In this section, using the results in Section 3, we determine the radio number of the Cartesian product of a level-wise regular tree and a complete graph.

It is well known that the center of tree T consists of one vertex r or two adjacent vertices r,r depending on diam(T) is even or odd. We view a tree T rooted at r or at both r,r respectively. Halász and Tuza [9] defined a level-wise regular tree to be a tree T in which all vertices at distance i from root r or {r,r} have the same degree, say, di for 0ih, where h (the largest distance from a vertex to the root) is the height of tree T. Note that a level-wise regular tree is determined by its center(s) and (d0,d1,,dh1). Denote the level-wise regular tree by Tz=Td0,d1,,dh1z, where z denotes the number of roots of the level-wise regular tree. A star K1,q is a tree consisting of q leaves and another vertex joined to all leaves by edges. A double star Dq is a graph obtained by joining the center vertices of two copies of star graph K1,q by an edge. A banana tree Bq,k is a graph constructed by connecting a single leaf from q distinct copies of a star K1,k with a single vertex that is distinct from all the vertices of the star graphs. Note that K1,q, Dq and Bq,k are level-wise regular trees Tq1, Tq+12 and Tq,2,k1, respectively.

Theirem 3.1. Let h1 and di,n3 for 0ih1 be integers. Denote ε=ε(Tz). Then rn(Tz◻Kn) is (15){[(i=1h1(2h2i1)(1ji(dj1)))+2h1]nd0+(2h+1)(n1),if z=1,[(i=0h1(2h2i1)(0ji(dj1)))+2h+1]2n2h1,if z=2.

Proof. The order of Tz, L(Tz) and diameter d of Tz are given by |V(Tz)|={1+d0+d0i=1h1(1ji(dj1)),if z=1,2+2i=0h1(0ji(dj1)),if z=2,

L(Tz)={d0+d0i=1h1(i+1)(1ji(dj1)),if z=1,2i=0h1(i+1)(0ji(dj1)),if z=2,

d={2h,if z=1,2h+1,if z=2. Substituting the above into (6), we obtain the right-hand side of (15) as a lower bound for rn(Tz◻Kn). We now prove that this lower bound is tight by giving an ordering of V(Tz◻Kn) which satisfies the conditions in Theorem 3.2. For this purpose, denote V(Kn)={v0,v1,,vn1} and V(Tz)={u0,u1,,um1} such that the ordering u0,u1,,un1 is obtained as follows.

Case 1. z=1.

In this case, let r be the unique center of T1. Denote d0 children of r by r0,,rd01. For 0id01, denote d11 children of ri by ri,0,ri,1,,ri,d11. Inductively, denote the dl1 children of ri1,i2,,il(0i1d01,0i2d11,,0ildl11) by ri1,i2,,il,il+1, where 0il+1dl1. Continue this process until all the vertices of T1 get indexed. Now obtain an ordering u0,u1,,um1 of vertices of T1 as follows: Set u0:=r and for 1tm1, set ut:=ri1,i2,,il, where j=1+i1+i2d0++ild0(d11)(dl1)+l+1thd0(d11)(dt1).

Case-2. z=2.

In this case, let r and r be two centers of T2. Denote d01 children of r and r by r0,r1,,rd02 and r0,r1,,rd02, respectively. For 0id02, denote d11 children of ri and ri by ri,0,ri,1,,ri,d11 and ri,0,ri,1,,ri,d11, respectively. Inductively, denote the dl1 children of ri1,i2,,il and ri1,i2,,il, where 0i1d02,0i2d11,,0ildl11 by ri1,i2,,il,il+1 and ri1,i2,,il,il+1 respectively, where 0il+1dl1. Continue this process until all the vertices of T2 are indexed. Rename xj:=ri1,i2,,il and xj:=ri1,i2,,il, where j=1+i1+i2(d01)++il(d01)(d11)(dl1)+l+1th(d01)(d11)(dt1). Now obtain an ordering u0,u1,,um1 as follows: Set u0:=r,um1:=r and for 1tm2, set ut:={xt/2,if t0 (mod 2),x(t+1)/2,if t1 (mod 2).

We now consider the following two cases to give an ordering w0,w1,,wmn1 of V(T◻Kn)={(ui,vj):i=0,1,,m1,j=0,1,,n1}.

Case 1. |W(T)|=1.

We consider the following two subcases to define an ordering w0,w1,,wmn1 of V(T1◻Kn).

Subcase 1. |V(T1)||V(Kn)|.

In this case, for 0tmn1, define wt:=(ui,vj), where i{0,1,,m1}j{0,1,,n1} and t:={m(ji)+iif ji and jin1,m(n+ji)+iif j<i and ij1,m(n+ji)+i1if j<i and ij=1,mn1if ji=n1.

Subcase 2. |V(T1)|>|V(Kn)|.

In this case, for xlcm(m,n)t(x+1)lcm(m,n)1, where 0xgcd(m,n)1, set αt:={(ui,vj+x),if ti (mod m), tj (mod n) and 0jn(x+1),(ui,vj+xn),if ti (mod m), tj (mod n) and nxjn1.

Define an ordering w0,w1,,wmn1 of V(T1◻Kn) as follows: For 0tmn1, set wt:={αt,if 0tmnm1,αt+1,if mnmtmn2,αmnm,if t=mn1.

Then, for above defined ordering, it is clear that (a) L(ui0)=L(uimn1)=L(u0)=0, (b) vjt and vjt+1 are distinct for 0tmn2, (c) uit and uit+1 are in different branches for 0tmn2, and (d) L(uit)d/2 for all 0tmn1. Hence the conditions (a)-(d) in Theorem 3.3 are satisfied. We now show that the condition (e) of Theorem 3.3 holds.

The above defined ordering w0,w1,,wmn1 of V(T1◻Kn) satisfies (13).

Let wa=(uia,vja) and wb=(uib,vjb)(0a<bmn1) be two vertices such that uia and uib are in the same branch when viewed as vertices of T1. Denote the right-hand side of (13) by Pa,b. In this case, note that d03, ε=1 and L(ut)d/2 for all 0tmn1. Observe that for wa=(uia,vja) and wb=(uib,vjb) such that uia and uib are in the same branch of T2, there are two possibilities: (1) iaib (2) ia=ib. In case (1), we have baαd0, where αϕ(uia,uib). Hence, Pa,b=((ba1)/2)(d+1)t=a+1b1L(uit)((ba1)/2)(d+1)t=a+1b1(d/2)=((ba1)/2)=α(d01)/2ϕ(uia,uib). In case (2), note that bamd+1. Hence, Pa,b=((ba1)/2)(d+1)t=a+1b1L(uit)((ba1)/2)(d+1)t=a+1b1(d/2)=((ba1)/2)=d/2ϕ(uia,uib).

Case 2. |W(T)|=2

Again we consider the following two subcases to define an ordering w0,w1,,wmn1 of V(T2◻Kn).

Subcase 1. |V(T)||V(Kn)|.

In this case, for 0tmn1, define wt:=(ui,vj), where i{0,1,,m1},j{0,1,,n1} and t:={m(ji)+i,if ji,m(n+ji)+i,if j<i.

Subcase 2. |V(T)|>|V(Kn)|.

In this case, define an ordering w0,w1,,wmn1 of V(T2◻Kn) as follows: For xlcm(m,n)t(x+1)lcm(m,n)1, where 0xgcd(m,n)1, set wt:={(ui,vj+x),if ti (mod m), tj (mod n) and 0jn(x+1),(ui,vj+xn),if ti (mod m), tj (mod n) and nxjn1,

Then, for above defined ordering, it is clear that (a) L(ui0)=L(uimn1)=L(u0)=0, (b) vjt and vjt+1 are distinct for 0tmn2, (c) uit and uit+1 are in different branches for 0tmn2, and (d) L(uit)(d1)/2 for all 0tmn1. Hence the conditions (a)-(d) of Theorem 3.3 are satisfied. Now we prove that the condition (e) of Theorem 3.3 holds.

Claim 2. The above defined ordering w0,w1,,wmn1 of V(T2◻Kn) satisfies (13).

Let wa=(uia,vja) and wb=(uib,vjb)(0a<bmn1) be two vertices such that uia and uib are in the same branch when viewed as vertices of T2. Denote the right-hand side of (13) by Pa,b. In this case, note that d03, ε=0 and L(ut)(d1)/2 for all 0tmn1. Observe that for wa=(uia,vja) and wb=(uib,vjb) such that uia and uib are in the same branch of T2, there are two possibilities: (1) iaib, (2) ia=ib. In case (1), we have baα(d01), where αϕ(uia,uib). Hence, Pa,b=((ba1)/2)dt=a+1b1L(uit)1/2((ba1)/2)dt=a+1b1((d1)/2)1/2=(ba2)/2=α(d01)/2ϕ(uia,uib). In case (2), note that bamd+1. Hence, Pa,b=((ba1)/2)dt=a+1b1L(uit)1/2((ba1)/2)dt=a+1b1((d1)/2)1/2=(ba2)/2=α(d1)/2ϕ(uia,uib)◻

Corollary 4.2. Let q3 and n4 be integers. Then rn(K1,q◻Kn)=(q+3)n3.
Corollary 4.3. Let q2 and n4 be integers. Then rn(Dq◻Kn)=2n(q+3)3.
Corollary 4.4. Let k,n4 and q5 be integers. Then rn(Bq,k◻Kn)=qn(k+6)+7(n1).
Exmple 4.5 . In Table 1, a vertex ordering O(V(K1,6◻K7)):=w0,w1,,w48 and an optimal radio labelling of K1,6◻K7 are shown.

Table 1 rn(K1,6◻K7)=60
(ui,vj) v0 v1 v2 v3 v4 v5 v6
u0 w00 w79 w1418 w2127 w2836 w3545 w4860
u1 w4253 w12 w811 w1520 w2229 w2938 w3647
u2 w3748 w4354 w23 w912 w1621 w2330 w3039
u3 w3140 w3849 w4455 w34 w1013 w1722 w2431
u4 w2532 w3241 w3950 w4556 w45 w1114 w1823
u5 w1924 w2633 w3342 w4051 w4657 w56 w1215
u6 w1316 w2025 w2734 w3443 w4152 w4758 w67
Example 4.6. In Table 2, a vertex ordering O(V(D5◻K7)):=w0,w1,,w83 and an optimal radio labelling of D5◻K7 are shown.
Table 2 rn(D5◻K7)=109
(ui,vj) v0 v1 v2 v3 v4 v5 v6
u0 w00 w3648 w7296 w2432 w6080 w1216 w4864
u1 w4966 w12 w3750 w7398 w2534 w6182 w1318
u2 w1419 w5067 w23 w3851 w7499 w2635 w6283
u3 w6384 w1520 w5168 w34 w3952 w75100 w2736
u4 w2837 w6485 w1621 w5269 w45 w4053 w76101
u5 w77102 w2938 w6586 w1722 w5370 w56 w4154
u6 w4255 w78103 w3039 w6687 w1823 w5471 w67
u7 w78 w4356 w79104 w3140 w6788 w1924 w5572
u8 w5673 w89 w4457 w80105 w3241 w6889 w2025
u9 w2126 w5774 w910 w4558 w81106 w3342 w6990
u10 w7091 w2227 w5875 w1011 w4659 w82107 w3443
u11 w3545 w7193 w2329 w5977 w1113 w4761 w83109

5. Concluding remarks

In [11], Kim et al. determined the radio number of a path Pm(m4) and the complete graph Kn(n3) as follows: (16)rn(Pm◻Kn)={m2n2m+22,if m is even,m2n2m+n+22,if m is odd. This result can be proved using our results. The outline of the proof is as follows: Since a path Pm is a tree, we use Theorems 3.1 and 3.2 to prove the result. Note that the diameter d of Pm is m1. We consider the following two cases.

Case 1. m is even. In this case, the total level of a path Pm is given by L(Pm)=m(m2)/2. Substituting this into (6), we obtain rn(Pm◻Kn)(m2n2m+2)/2. Now observe that the ordering of V(Pm◻Kn) given in [11] satisfies the conditions (a)-(c) in Theorem 3.2 and hence we obtain that the first line in (16) holds.

Case 2. m is odd. In this case, the total level of a path Pm is given by L(Pm)=(m21)/2. Substituting this into (6), we obtain rn(Pm◻Kn)(m2n2m+n)/2. Now assume that rn(Pm◻Kn)=(m2n2m+n)/2. Then by Theorem 3.2, there exists an ordering w0,w1,,wmn1 of V(Pm◻Kn) such that the conditions (a)-(c) in Theorem 3.2 hold. Since L(ui0)=L(uimn1) by (a), observe that there exists a vertex wk=(uik,vjk) such that L(uik)=(m1)/2 and L(uik1)0,L(uik+1)0. Note that d(uik1,uik+1)=L(uik1)+L(uik+1)2ϕ(uik1,uik+1) with ϕ(uik1,uik+1)1. Denote the right-hand side of (9) by Sa,b and consider uik1 and uik+1 in (9). Then we obtain Sk1,k+1=L(uik1)+2L(uik)+L(uik+1)(d+1)=L(uik)+2((d+1)/2)+L(uik+1)(d+1)=L(uik1)+L(uik+1)>L(uik1)+L(uik+1)2ϕ(uik1,uik+1)=d(uik1,uik+1), which is a contradiction. Hence, rn(Pm◻Kn)(m2n2m+n+2)/2. Now observe that the ordering of V(Pm◻Kn) given in [11] satisfies the conditions (a)-(c) in Theorem 3.2. Hence we obtain that the second line in (16) holds, and this completes the proof.

Acknowledgement

The authors are grateful to the two anonymous referees for their careful reading of the manuscript and helpful comments. The research of the second author is supported by Research Promotion under Technical Education – STEM research project grant (Project ID: 202122MATH01) of the Government of Gujarat.

Statements and declarations

The authors have no relevant financial or non-financial interests to disclose.

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