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 -labelling of a graph is a function
such that if
and if . The span of is defined as , and
the -number (or the
-number) of is the minimum span of an -labelling of . The -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
by (the diameter of a
graph is ). In [7,6], Chartrand et
al. introduced the concept of the radio labelling problem by
extending the condition on distance in -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
is a mapping
such that the following hold for every pair of distinct vertices
of
,
The integer
is called the
label of
under
, and the
span of
is defined as
. The
radio number of
is defined as
with minimum taken over all radio labellings
of
. A radio labelling
of
is called
optimal if
.
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
to some vertex so that the span of a radio labelling is equal to the
maximum label used. Since , any radio labelling always assigns different labels
to distinct vertices. Therefore, a radio labelling of graph with order induces the linear order of the vertices of
such that A linear order of is called an optimal linear order if
it induced by some optimal radio labelling of .
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 and diameter . 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 -ary trees of
height , for . 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 , the neighbourhood of any is . The distance between two vertices and in , denoted by , is the length of the shortest
path joining and in . The diameter of a graph , denoted by , is . We drop the suffix in above-defined terms when is clear in the context. A complete
graph is a graph on vertices in which any two vertices are
adjacent. A tree is a connected acyclic graph. We fix for
tree of order and throughout the paper. A vertex
is an internal
vertex of if it has degree
greater than one and is a leaf otherwise. In [3,13], the
weight of from is defined as and the weight of is defined as A
vertex is a weight
center [3,13]
of if . Denote by the set of weight centers of . In [13], the following is proved about .
Lemma 2.1 .
[
13]
] If is a
weight center of a tree . Then
each component of contains at
most vertices.
Lemma 2.2.
[
13]
] Every tree has one or two weight centers, and
has two weight centers, say,
if and only if
is an edge of and consists of two equal sized
components.
A vertex is called [3,13] an
ancestor of a vertex , or
is a descendent of , if is on the unique path joining a weight
center and . Let be a vertex
adjacent to a weight center . The
subtree of induced by and all its descendants is called the
branch of at . Two vertices of 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
rooted at its weight center :
if , then is rooted at ; if , then is
rooted at and in the sense that both and are at level 0. In either case,
for each , define to indicate the level of in , and define
the total level of . For
any , define
Lemma 2.3 ([
3, Lemma 2.1]).
Let
be a tree with diameter
. Then for any
the following hold:
(a) ;
(b) if
and only if and are in different or opposite
branches;
(c) if
and only if has two weight
centers and and are in opposite branches;
(d) the distance
in between and can be expressed as
Define
Let and be two graphs. The
Cartesian product of and is the graph with such that
two vertices and are adjacent if and or and . It is clear from the
definition that .
Observation 2.4.
For a tree
of order
and a complete graph
,
(a) , and
(b) .
Let be an ordering of , where . Then each is an ordered
pair with and . Hence each vertex
appears
times and each vertex appears times in . For ,
and are called
distinct if .
For any ,
the distance between and is given by
3. A tight lower bound for
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 be a tree of order and diameter . Denote . Then
Proof. It is enough to prove that any radio labelling of
has no span less than
that of the right-hand side of (6).
Suppose is any radio labelling of
which induces an
ordering such that . Denote . By the definition of a radio labelling, for . Since by Observation 2.4, we have
for . Summing up
these inequalities, we obtain
Case 1. . In
this case, by the definition of , and
for all . Hence, using (5), we
obtain Substituting this into (7), we
have .
Case 2. . In
this case, , and
for all . Hence, using (5), we
obtain Substituting this into (7), we
have . 
Theorem 3.2.
Let
be a tree of order
and diameter
. Denote
. Then
if and
only if there exists an ordering
of
such that the following
hold:
(a) ;
(b) and are distinct for all ;
(c) for , the
distance between any two vertices and satisfies
Moreover, under these conditions (a)-(c), the mapping defined by is an optimal radio labelling of .
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 of induced by satisfies the followings: (1) , (2) and are in different branches
when and in opposite
branches when for all
, (3) and are distinct for all . Hence in this case
the definition of radio labelling
can be written as and
for , where . Summing
this last equality for to , we
obtain
Since is a radio labelling of
, we have . Also note that by (5). Substituting these into the
above equation, we obtain
Sufficiency. Suppose there exists an ordering (, , , ) of satisfies the conditions
(a)-(c) and is defined by (10) and (11). Note
that it is enough to prove that
is a radio labelling of
and the span of is the right-hand
side of (8). Denote . Let
and be two
arbitrary vertices. The span of is 
Theorem 3.3.
Let
be a tree of order
and diameter
. Denote
. Then
if and
only if there exists an ordering
of
with
such that the following all hold:
(a) ;
(b) and are distinct for ;
(c) and are in different branches
when and in opposite
branches when for ;
(d) when and
when for all ;
(e) for any such that
and are in the same branch
of , and satisfy
Proof. Necessity. Suppose (12) holds.
Then there exists an ordering (,
, , ) of such that the conditions
(a)-(c) of Theorem 3.2 holds. Hence the conditions (a)
and (b) are satisfied. Taking
and in (9), we
obtain . Hence we have for all as and .
Therefore, by Lemma 2.3, and are in different branches
when and in opposite
branches when for all
. Hence the
condition (c) is satisfied. Since , it is clear that for . For , considering and in (9) and
using (5), we obtain
In above equation, observe that when then by the
definition of and when then and are in different or in the
same branch of and hence . Thus
we have
Hence, the condition (d) is satisfied. To prove (e), let and be such that
and are in the same branch
of . Then . Using (5) in (9), (13) can be obtained easily from (9).
Sufficiency. Suppose there exists an ordering of such that conditions
(a)-(e) hold. To prove (12) holds, we show that an ordering
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 . Let be any two vertices. If and are in opposite branches, then
. Hence, . If and are in different branches, then
and .
If , then note that for all and hence . If , then note that and for all and hence . If and are in the same branch, then (9) can be obtained easily using (13). This completes the proof. 
Theorem 3.4.
Let
be a tree of order
and diameter
. Denote
. Then
if there
exists an ordering
of
such that
(a) ,
(b) and are distinct for all ,
(c) and are in different branches
when and in opposite
branches when for all
,
and one of the following holds:
(d) for all ;
(e) for all ;
(f) for all , when and
when and, if and such that
and are in the same branch
of then .
Proof. We show that if (a)-(c) and one of (d)-(f) holds for
an ordering
such that , 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 . We consider the following two
cases.
Case 1. . In
this case, recall that and for all by the definition of .
Subcase 1.1. Suppose (a)-(d) hold. It is clear that and for , . Let and be two arbitrary
vertices such that and
are in the same branch of
. If , then . Assume . If ,
then . If ,
then and hence . Therefore, . Assume . Then either or
. Without loss of generality, we may assume that . Then
.
Hence, .
Subcase 1.2. Suppose (a)-(c) and (e) hold. Note that . Since and for , we obtain for all .
Let and
be two arbitrary vertices such that and are in the same branch of . If , then . Assume . Note that
and hence . Therefore, .
Subcase 1.3. Suppose (a)-(c) and (f) hold. Assume is even. Then for all and hence . Assume is odd. Then note that only for one
vertex , from consecutive
vertices and for all other
vertices , . Hence, .
Case 2. . In
this case, recall that and for all
by the
definition of the ordering of .
Subcase 2.1. Suppose (a)-(d) hold. It is clear that and for , . Let and be two arbitrary
vertices such that and
are in the same branch of
. If , then . Assume . If , then . If , then
and hence . Therefore, . Assume . Then either or
. Without loss of generality, we may assume that . Then . Hence, .
Subcase 2.2. Suppose (a)-(c) and (e) hold. Note that . Since and for , we obtain for all .
Let and
be two arbitrary vertices such that and are in the same branch of . If , then . Assume that . Then note that
and hence . Hence, .
Subcase 2.3. Suppose (a)-(c) and (f) hold. Assume is even. Then for all and hence . Assume is odd. Then note that only for one
vertex , from consecutive vertices and for all
other vertices , . Hence, . 
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 consists of one vertex or two adjacent vertices depending on is even or odd. We view a
tree rooted at or at both respectively. Halász and Tuza
[9] defined a
level-wise regular tree to be a tree in which all vertices at distance from root or have the same degree, say,
for , where (the largest distance from a vertex to
the root) is the height of tree .
Note that a level-wise regular tree is determined by its center(s) and
. Denote
the level-wise regular tree by , where denotes the number of roots of the
level-wise regular tree. A star is a tree consisting of leaves and another vertex joined to all
leaves by edges. A double star is a graph obtained by joining the
center vertices of two copies of star graph by an edge. A banana
tree is a graph
constructed by connecting a single leaf from distinct copies of a star with a single vertex that is
distinct from all the vertices of the star graphs. Note that , and are level-wise regular trees
, and , respectively.
Theirem 3.1.
Let and for be integers. Denote
. Then
is
Proof. The order of , and diameter of are given by
Substituting the above into (6), we
obtain the right-hand side of (15) as
a lower bound for . We now prove that this lower bound is tight by giving an
ordering of which
satisfies the conditions in Theorem 3.2. For
this purpose, denote and such
that the ordering is obtained as
follows.
Case 1. .
In this case, let be the
unique center of . Denote children of by . For , denote children of by .
Inductively, denote the
children of by , where
. Continue
this process until all the vertices of get indexed. Now obtain an ordering
of vertices
of as follows: Set and for , set , where
.
Case-2. .
In this case, let and be two centers of . Denote children of and by and ,
respectively. For , denote
children of and by and
,
respectively. Inductively, denote the children of and , where by and
respectively, where . Continue this process until all the vertices of are indexed. Rename , where . Now obtain an
ordering as
follows: Set and for , set
We now consider the following two cases to give an ordering of .
Case 1. .
We consider the following two subcases to define an ordering of .
Subcase 1. .
In this case, for , define , where and
Subcase 2. .
In this case, for , where , set
Define an ordering of as follows: For , set
Then, for above defined ordering, it is clear that (a) , (b) and are distinct for , (c) and are in different branches for
, and (d) for all . 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 of satisfies (13).
Let and
be two vertices such that and are in the same branch when
viewed as vertices of . Denote
the right-hand side of (13) by . In this case, note that , and for all . Observe that for
and such that and are in the same branch of , there are two possibilities: (1)
(2) . In case (1), we have , where . Hence,
. In case (2), note that . Hence, .
Case 2.
Again we consider the following two subcases to define an ordering
of .
Subcase 1. .
In this case, for , define , where and
Subcase 2. .
In this case, define an ordering of as follows: For , where , set
Then, for above defined ordering, it is clear that (a) , (b) and are distinct for , (c) and are in different branches for
, and (d) for all . 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 of satisfies (13).
Let and
be two vertices such that and are in the same branch when
viewed as vertices of . Denote
the right-hand side of (13) by . In this case, note that , and for all . Observe that for
and such that and are in the same branch of , there are two possibilities: (1)
, (2) . In case (1), we have , where . Hence,
. In case
(2), note that .
Hence, . 
Corollary 4.2.
Let and be integers. Then
Corollary 4.3.
Let and be integers. Then
Corollary 4.4.
Let and be integers. Then
Exmple 4.5 .
In Table 1, a vertex ordering
and an optimal radio labelling of
are shown.
Example 4.6.
In Table 2, a vertex ordering
and an optimal radio labelling of are shown.
In [11], Kim et
al. determined the radio number of a path and the complete graph
as follows: This result can be proved using our results. The
outline of the proof is as follows: Since a path is a tree, we use Theorems 3.1 and 3.2 to
prove the result. Note that the diameter of is . We consider the following two
cases.
Case 1. is even. In
this case, the total level of a path is given by . Substituting this into
(6), we obtain . Now observe that the ordering of 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. is odd. In
this case, the total level of a path is given by . Substituting this
into (6), we obtain . Now assume that . Then by Theorem 3.2, there exists an ordering of such that the conditions
(a)-(c) in Theorem 3.2 hold. Since by (a),
observe that there exists a vertex such that and . Note that
with . Denote the right-hand side of (9) by
and consider and in (9). Then
we obtain , which is a contradiction. Hence,
. Now observe that the ordering of 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.