Upper bound for radio k -chromatic number of graphs in connection with partition of vertex set

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Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set Laxman Saha Department of Mathematics, Balurghat College, Balurghat 733101, India Received 20 January 2019; received in revised form 23 March 2019; accepted 25 March 2019 Available online xxxx

Abstract For a simple connected graph G = (V (G), E(G)) and a positive integer k with 1 ⩽ k ⩽ diam(G), a radio k-coloring of G is a mapping f : V (G) → {0, 1, 2, . . .} such that | f (u) − f (v)| ⩾ k + 1 − d(u, v) holds for each pair of distinct vertices u and 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 span of a radio k-coloring f is the number max f (u). The main aim of the radio k-coloring problem is to determine the minimum span of a u∈V (G)

radio k-coloring of G, denoted by r ck (G). Due to NP-hardness of this problem, people are finding lower bounds for r ck (.) for particular families of graphs or general graphs G. In this article, we concentrate on finding upper bounds of radio k-chromatic number for general graphs and in consequence a coloring scheme depending on a partition of the vertex set V (G). We apply these bounds to cycle graph Cn and hypercube Q n and show that it is sharp when k is close to the diameter of these graphs. c 2019 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND ⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Keywords: Channel assignment problem; Radio k-coloring; Radio k-chromatic number; Span

1. Introduction Due to rapid growth in the use of wireless communication services and the corresponding scarcity and the high cost of radio spectrum bandwidth, channel assignment problem (CAP) is becoming highly important. Radio k-coloring of a graph is a variation of channel assignment problem introduced by Hale [1]. The channel assignment problem is the problem of assigning channels (non-negative integers) to the stations in a way which avoids interference and uses the spectrum as efficiently as possible. In order to avoid interference, the separation between the channels assigned to a pair of near-by stations must be large enough; the amount of the required separation depends on the distance between the two stations. When the distance between two stations is smaller, the difference in the channel assigned must be relatively large, whereas two stations at a large distance may be assigned channels with a small difference. A variation of the channel assignment problem is the L( j, k)-labeling (coloring ) problem of graphs. The L( j, k)-labeling problem ( j, k ⩾ 0), and its variations have been studied extensively. A major concern of this problem is to seek an assignment of labels (which are non-negative integers) to the vertices of a graph such that Peer review under responsibility of Kalasalingam University. E-mail address: [email protected]. https://doi.org/10.1016/j.akcej.2019.03.024 c 2019 Kalasalingam University. Production and Hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND 0972-8600/⃝ license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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the span (difference) between the largest and smallest labels used is minimized, subject to that adjacent vertices receive labels with separation at least j and vertices at distance two apart receive labels with separation at least k. Motivated by FM channel assignments, the radio k-coloring problem was introduced in [2] by Chartrand et al. Let G be a simple connected graph with diameter diam(G). For a positive integer k with 1 ⩽ k ⩽ diam(G), a radio k-coloring f of G is an assignment of non-negative integers to the vertices of G such that for every two distinct vertices u and v of G | f (u) − f (v)| ⩾ k + 1 − d(u, v).

(1)

The span of a radio k-coloring f , denoted by span( f ), is defined as maxv∈V (G) f (v) and the radio k-chromatic number of G, denoted by r ck (G), is min f { span( f )} where the minimum is taken over all radio k-colorings of G. For k = diam(G), the number r ck (G) is known as radio number of G and is denoted by r n(G). In this paper we refer (1) as the radio condition. A radio k-coloring f of G is called minimal if span( f ) = r ck (G). Without loss of generality for a minimal radio k-coloring f we assume that minv∈V (G) f (v) = 0, otherwise the span of f can be reduced further by subtracting the integer minv∈V (G) f (v) from all the labels of the vertices of the graph. The problem of finding the radio k-chromatic number of a graph is of great interest for its widespread applications to channel assignment problem. So far, radio number is known only for very limited families of graphs and specific values of k. For paths and cycles the radio numbers were studied by Chartrand et al. [2–4] and the exact values of the radio number remained open until solved by Liu and Zhu [5]. The radio number of any hypercube was determined by Khennoufa et al. [6] and Kola et al. [7], independently. Generalized prism graphs, denoted Z n,s , s ⩾ 1, n ⩾ s, have vertex set⌊{(i, ⌋j) : i = 1, 2 and ⌊ s−1j⌋= 1, . . . , n} and edge set {((i, j), (i, j ± 1))} ∪ {((1, i), (2, i + σ )) : σ ∈ S}, where S = {− s−1 , . . . , 0, . . . }. Ortiz et al. [8] have studied the radio number of generalized prism graphs 2 2 and have computed the exact value of radio number for some specific types of generalized prism graphs. The radio number of complete m-ary tree was determined by Li et al. [9]. For two positive integers m ⩾ 3 and n ⩾ 3, the Toroidal grids Tm,n are the cartesian product Cm □Cn . Morris et al. [10] have determined the radio number of Tn,n and Saha et al. [11] have given exact value for radio number of Tm,n when mn ≡ 0 (mod 2). The radio numbers of the square of paths and cycles were studied in [12,13]. It is not easy to find radio k-chromatic number for all graphs as the problem is NP-hard. So researchers are finding lower and upper bounds of radio k-chromatic number for some class of graphs and specific values of k. Recently, lower bounds of radio k-chromatic number for general graph have been studied in [14,15]. In literature, there is no upper bound of radio k-chromatic number for general graphs. Thus one major task is to give an optimal upper bound for r ck (G). In this article, we give a coloring scheme to find an upper bound for radio k-chromatic number r ck (G) of a given graph G. This coloring scheme works for general graphs and it depends on a partition of the vertex set V (G) into two partite sets satisfying some conditions. Also, we apply these bounds to cycle graphs Cn and hypercubes Q n and show that it is sharp when k is close to the diameter of these graphs. Rest of the paper is organized as follows: In Section 2, we give an upper bound of radio k-chromatic number for general graph G when k is the diameter of G. Section 3 deals with the same when 2 ⩽ k ⩽ diam(G). In Section 4, we apply these results to particular families of graphs. Section 5 contains a concluding remark. 2. Upper bound for radio number First we find an upper bound for the radio number r n(G) of a simple connected graph G with even number of vertices. For this we need the following proposition that will ensure the existence of a partition of the vertex set V (G) into two sets satisfying some conditions. Proposition 2.1. Let G be a simple connected graph with 2m vertices. Then for every partition of the vertex set V (G) into two sets U1 = {u i : 0 ⩽ i ⩽ m − 1} and U2 = {u i′ : 0 ⩽ i ⩽ m − 1} there exist non-negative integers di , 0 ⩽ i ⩽ m − 2 such that ′ (a) d(u i , u i+1 ), d(u i′ , u i+1 ) ⩾ di , 0 ⩽ i ⩽ m − 2; ′ (b) d(u i , u i+1 ) ⩾ di − 1, 0 ⩽ i ⩽ m − 2; (c) d(u i+1 , u i′ ) ⩾ di + 1, 0 ⩽ i ⩽ m − 2.

Proof. If d(u j , u j+1 ) = a j , d(u ′j , u ′j+1 ) = b j , d(u j , u ′j+1 ) = c j and d(u j+1 , u ′j ) = e j then we take d j = min{a j , b j , c j + 1, e j − 1}, 0 ⩽ j ⩽ m − 2. □ Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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L. Saha / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx Table 1 Comparison between the values of diam(G) + 1 − d(u, v) and | f (u) − f (v)|. Pair of vertices u, v ∈ U1 ∪ U2

j −i

d(u, v)

diam(G) + 1 − d(u, v)

| f (u) − f (v)|

u, v ∈ U1 or u, v ∈ U2

1 ⩾2

⩾ di ⩾1

⩽ diam(G) + 1 − di ⩽ diam(G)

⩾ ai ⩾ diam(G) + 1 − di ⩾ ai + ai+1 ⩾ diam(G) + 1

u ∈ U1 , v ∈ U2

0 1 ⩾2

⩾ li ⩾ di − 1 ⩾1

⩽ diam(G) + 1 − li ⩽ diam(G) + 2 − di ⩽ diam(G)

⩾ diam(G) + 1 − li ⩾ ai + 1 ⩾ diam(G) + 2 − di ⩾ ai + ai+1 + 1 ⩾ diam(G) + 2

u ∈ U2 , v ∈ U1

1 ⩾2

⩾ di + 1 ⩾1

⩽ diam(G) − di ⩽ diam(G)

⩾ ai − 1 ⩾ diam(G) − di ⩾ ai + ai+1 − 1 ⩾ diam(G)

The following theorem gives an upper bound for radio number of graphs with even order. Theorem 2.1. Let G be a simple connected graph with 2m vertices and diameter diam(G). For a partition of the vertex set V (G) into two sets U1 = {u i : 0 ⩽ i ⩽ m − 1} and U2 = {u i′ : 0 ⩽ i ⩽ m − 1}, if di and l j for 0 ⩽ i ⩽ m − 2 and 0 ⩽ j ⩽ m − 1 are non-negative integers such that ′ d(u i , u i+1 ), d(u i′ , u i+1 ) ⩾ di , 0 ⩽ i ⩽ m − 2; ′ d(u i , u i+1 ) ⩾ di − 1, 0 ⩽ i ⩽ m − 2; d(u i+1 , u i′ ) ⩾ di + 1, 0 ⩽ i ⩽ m − 2; d(u i , u i′ ) = li , 0 ⩽ i ⩽ m − 1; ∑m−2 ∑m−1 then r n(G) ⩽ i=0 ai + i=0 (diam(G) − li ) + 1, where a0 , a1 , . . . , am−2 are given by ⎧ + 1 −⌉di , if 2di ⩽ diam(G) + 1, ⎨diam(G) ⌈ diam(G) + 1 ai = , if 2di > diam(G) + 1. ⎩ 2

(a) (b) (c) (d)

Proof. If 2di > diam(G)+1 then ai =

⌈

diam(G)+1 2

⌉

and diam(G)+1−di < diam(G)+1−

⌊

diam(G)+1 2

⌋

=

⌈

diam(G)+1 2

⌉ .

So ai ⩾ diam(G) + 1 − di for 0 ⩽ i ⩽ m − 2. Next we show that ai + ai+1 − 1 ⩾ diam(G) for 0 ⩽ i ⩽ m − 3. If both 2di and 2di+1 are greater than diam(G) + 1 then the inequality is obviously true. If both 2di and 2di+1 are less than or equal to diam(G) + 1 then ai + ai+1 = 2(diam(G) + 1) − di − di+1 ⩾ diam(G) + 1.⌋Finally, ⌊ ⌈ if one⌉of 2di and 2di+1 diam(G)+1 is greater than or equal to diam(G) + 1 then ai + ai+1 ⩾ diam(G) + 1 − diam(G)+1 + = diam(G) + 1. 2 2 Thus ai + ai+1 − 1 ⩾ diam(G) for 0 ⩽ i ⩽ m − 2. Now we define a coloring f of G as follows. For this we take a−1 = 0 and l−1 = diam(G). f (u i ) =

i i ∑ ∑ ( ) ( ) a j−1 + diam(G) − l j−1 and f (u i′ ) = a j−1 + diam(G) − l j + 1, 0 ⩽ i ⩽ m − 1. j=0

j=0

In Table 1, we check the radio condition for f . Without loss of generality, let i ⩽ j, and for u, v ∈ U1 ∪ U2 we take u ∈ {u i , u i′ } and v ∈ {u j , u ′j }. Since ai ⩾ diam(G) + 1 − di for 0 ⩽ i ⩽ m − 1 and ai + ai+1 ⩾ diam(G) + 1 for 0 ⩽ i ⩽ m − 2, we obtain the following table. From this table it is clear that | f (u) − f (v)| ⩾ diam(G) + 1 − d(u, v) ∑m−2 ∑m−1 for all u, v ∈ V . Therefore f is a radio coloring of G and the span of f is i=0 ai + i=0 (diam(G) − li ) + 1. □ Example 2.1. We illustrate the above theorem considering the circulant graph Ci 10 (1, 2) in Fig. 1. The diameter of Ci 10 (1, 2) is 3. If we take a partition of the vertex set of Ci 10 (1, 2) into two partite sets U1 = {x0 , x2 , x4 , x6 , x8 } and U2 = {x1 , x3 , x5 , x7 , x9 } then [d0 , d1 , d2 , d3 ] = [1, 1, 1, 1]; [l0 , l1 , l2 , l3 , l4 ] = [3, 3, 3, 3, 3] and [a0 , a1 , a2 , a3 ] = ∑3 ∑4 [3, 3, 3, 3] (see Fig. 2). Thus r n(Ci 10 (1, 2)) ⩽ i=0 ai + i=0 (3−li )+1 = 13. One can show that r n(Ci 10 (1, 2)) = 13. Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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Fig. 1. The circulant graph Ci 10 (1, 2).

Fig. 2. Partition of V (Ci 10 (1, 2)) with di and li .

In the theorem below we give an upper bound to the radio number of graphs with odd order. Theorem 2.2. Let G be a simple connected graph with 2m + 1 vertices and diameter diam(G). For a partition of V (G) − {u}, where u is a vertex in G, into two sets U1 = {u i : 0 ⩽ i ⩽ m − 1} and U2 = {u i′ : 0 ⩽ i ⩽ m − 1}, if di and li , 0 ⩽ i ⩽ m − 1, are non-negative integers such that ′ d(u i , u i+1 ), d(u i′ , u i+1 ) ⩾ di , 0 ⩽ i ⩽ m − 2; ′ d(u i , u i+1 ) ⩾ di − 1, 0 ⩽ i ⩽ m − 2; d(u i+1 , u i′ ) ⩾ di + 1, 0 ⩽ i ⩽ m − 2; d(u i , u i′ ) = li , 0 ⩽ i ⩽ m − 1; dm−1 = min{d(u, u 0 ), d(u, u ′0 )} or min{d(u, u ′m−1 ) − 1, d(u, u m−1 )} according as d(u, u 0 ) ⩾ d(u, u ′m−1 ) or d(u, u 0 ) < d(u, u ′m−1 ), respectively; ∑m−1 then r n(G) ⩽ i=0 (ai + diam(G) − li ), where a0 , a1 , . . . , am−1 are given by ⎧ + 1 −⌉di , if 2di ⩽ diam(G) + 1, ⎨diam(G) ⌈ diam(G) + 1 ai = , if 2di > diam(G) + 1. ⎩ 2

(a) (b) (c) (d) (e)

Proof. We give the proof of this theorem by considering the following two cases. ′ Case I: In this case we take ⌈dm−1 = min{d(u, u 0 ), d(u, u ′0 )}. Then d(u, u 0 ) ⩾ d⌊m−1 and d(u, ⌉ ⌋ u⌈0 ) ⩾ dm−1 ⌉ . If diam(G)+1 diam(G)+1 diam(G)+1 2di > diam(G) + 1 then ai = and diam(G) + 1 − di < diam(G) + 1 − = . So 2 2 2 ai ⩾ diam(G) + 1 − di for 0 ⩽ i ⩽ m − 1. Next we show that ai + ai+1 − 1 ⩾ diam(G) for 0 ⩽ i ⩽ m − 2. If both 2di and 2di+1 are greater than diam(G) + 1 then the inequality is obviously true. If both 2di and 2di+1 are less than or equal to diam(G) + 1 then ai + ai+1 = 2(diam(G) + 1) − di − di+1 ⩾ diam(G) + 1.⌋Finally, ⌊ ⌈ if one⌉of 2di and 2di+1 diam(G)+1 is greater than or equal to diam(G) + 1 then ai + ai+1 ⩾ diam(G) + 1 − + diam(G)+1 = diam(G) + 1. 2 2 Thus ai + ai+1 − 1 ⩾ diam(G) for 0 ⩽ i ⩽ m − 2. Let us take a−1 = am−1 and l−1 = diam(G). We define a coloring Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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Table 2 Comparison between the values of diam(G) + 1 − d(v, w) and | f (v) − f (w)|. Pair of vertices v, w ∈ U1 ∪ U2

j −i

d(v, w)

diam(G) + 1 − d(u, v)

| f (v) − f (w)|

v, w ∈ U1 or v, w ∈ U2

1 ⩾2

⩾ di ⩾1

⩽ diam(G) + 1 − di ⩽ diam(G)

⩾ ai ⩾ diam(G) + 1 − di ⩾ ai + ai+1 ⩾ diam(G) + 1

v ∈ U1 , w ∈ U2

0 1 ⩾2

⩾ li ⩾ di − 1 ⩾1

⩽ diam(G) + 1 − li ⩽ diam(G) + 2 − di ⩽ diam(G)

⩾ diam(G) + 1 − li ⩾ ai + 1 ⩾ diam(G) + 2 − di ⩾ ai + ai+1 + 1 ⩾ diam(G) + 2

v ∈ U2 , w ∈ U1

1 ⩾2

⩾ di + 1 ⩾1

⩽ diam(G) − di ⩽ diam(G)

⩾ ai − 1 ⩾ diam(G) − di ⩾ ai + ai+1 − 1 ⩾ diam(G)

f of G as follows. For 0 ⩽ i ⩽ m − 1, f (u) = 0, i ∑ (a j−1 + diam(G) − l j−1 ) f (u i ) = j=0 i ∑ ′ f (u i ) = (a j−1 + diam(G) − l j ) + 1. j=0

In Table 2, we check the radio condition for f . Without loss of generality we take i ⩽ j and for v, w ∈ U1 ∪ U2 we take v ∈ {u i , u i′ } and w ∈ {u j , u ′j }. Since ai ⩾ diam(G) + 1 − di for 0 ⩽ i ⩽ m − 1 and ai + ai+1 ⩾ diam(G) + 1 for 0 ⩽ i ⩽ m − 2, we obtain Table 2. From this table it is clear that | f (v) − f (w)| ⩾ diam(G) + 1 − d(v, w) for all v, w ∈ V − {u}. Now, let w ∈ V − {u}. If w = u 0 then f (w) − f (u) = a−1 + diam(G) − l−1 = am−1 ⩾ diam(G) + 1 − dm−1 and if w = u ′0 then f (w) − f (u) = a−1 + diam(G) − l0 + 1 ⩾ am−1 + 1 ⩾ diam(G) + 1 − dm−1 . Again if w ∈ V − {u, u 0 , u ′0 } then | f (u) − f (w)| ⩾ diam(G). So | f (v) − f (w)| ⩾ diam(G) + 1 − d(v, w) for all v, w ∈ V . Case II: Here we take dm−1 = min{d(u, u ′m−1 ) − 1, d(u, u m−1 )}. Then we get, d(u, u m−1 ) ⩾ dm−1 and d(u, u ′m−1 ) ⩾ dm−1 + 1. Suppose a−1 = 0, l−1 = diam(G). We define a coloring f of G as follows. For 0 ⩽ i ⩽ m − 1, f (u i ) =

i ∑ (a j−1 + diam(G) − l j−1 ) j=0

i ∑ f (u i′ ) = (a j−1 + diam(G) − l j ) + 1 j=0 m ∑ f (u) = (a j−1 + diam(G) − l j−1 ). j=0

Similarly as in Case I we can show that | f (v) − f (w)| ⩾ diam(G) + 1 − d(v, w), for all v, w ∈ V − {u}. Now, let w ∈ V −{u}. If w = u m−1 then f (u)− f (w) = am−1 +diam(G)−lm−1 ⩾ am−1 ⩾ diam(G)+1−dm−1 and if w = u ′m−1 then f (u) − f (w) = am−1 − 1 ⩾ diam(G) − dm−1 ⩾ diam(G) + 1 − d(u, w). Again if w ∈ V − {u, u m−1 , u ′m−1 } then | f (u) − f (w)| ⩾ am−1 + am−2 − 1 ⩾ diam(G). So | f (v) − f (w)| ⩾ diam(G) + 1 − d(v, w) for all v, w ∈ V . ∑m−1 In the both cases the span of f is i=0 (ai + diam(G) − li ). □ Remark 2.1. Theorem 2.1 (respectively Theorem 2.2 ) holds for every partition of the vertex set of the graph G with suitable di ’s and li ’s. To get a better upper bound of the radio number of G we have to take that partition of V (G) ∑m−2 ∑m−1 ∑m−1 and those integers di and li for which i=0 ai + i=0 (diam(G) − li ) + 1 (respectively i=0 (ai + diam(G) − li )) is minimized. Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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Definition 2.1. The triameter of a graph G = (V (G), E(G)) is defined as the smallest positive integer M such that d(u, v) + d(v, w) + d(w, u) ⩽ M for every triplet u, v and w in V (G). The following lemma presents a relationship between the diameter and the triameter of a simple connected graph. Lemma 2.1. Let M and diam(G) be the triameter and diameter of a simple connected graph G. Then we have the inequality 2 diam(G) ⩽ M ⩽ 3 diam(G). Proof. Since max{d(u, v) + d(v, w) + d(w, u) : for all u, v, w ∈ V (G)} ⩽ 3 diam(G) and M is the smallest integer such that d(u, v) + d(v, w) + d(w, u) ⩽ M for all u, v, w ∈ V (G), we get M ⩽ 3 diam(G). If the vertices u and v are chosen in such a way that d(u, v) = diam(G), then from the triangle inequality d(v, w) + d(w, u) ⩾ d(u, v) = diam(G). Therefore d(u, v) + d(v, w) + d(w, u) ⩾ 2 diam(G). □ In [14], Saha and Panigrahi have given lower bounds of radio k-chromatic number for general graphs. In the following two theorems we stated these lower bounds presented in [14]. Theorem 2.3. Let G be an n-vertex simple connected graph with diameter diam(G). Then ⎧⌈ ⌉( ) 3(diam(G) + 1) − M n−2 ⎪ ⎪ + 1, if n is even, ⎪ ⎪ ⎨ 2 2 r n(G) ⩾ ⌈ ⌉( ) ⎪ ⎪ 3(diam(G) + 1) − M n−1 ⎪ ⎪ , if n is odd. ⎩ 2 2 where M is the triameter of G. Theorem 2.4. For an n-vertex simple connected graph G with diameter diam(G), ⌈ ⌉⌊ ⌋ ⌈ ⌉ 3(k + 1) − M n−1 M −2 r ck (G) ⩾ , ⩽ k ⩽ diam(G) − 1 2 2 2 where M is the triameter of G. Definition 2.2. An even order simple connected graph G with diameter diam(G) is said to have an antipodal perfect matching if there exists a partition of V (G) into two sets, U1 = {u i : 0 ⩽ i ⩽ n2 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ n2 − 1}, such that d(u i , u i′ ) = diam(G), 0 ⩽ i ⩽ n2 − 1. For an odd order graph G with diameter diam(G) is said to have an antipodal perfect matching if there exists a partition of V (G) \ {u}, for some vertex u, into two sets as above. We denote the collection of all simple connected graphs having an antipodal perfect matching by Ω . Example 2.2. The cycle graph Cn , hypercube Q n all belong to the class Ω . Also it is clear that if a graph G ∈ Ω then its power graph G r ∈ Ω . Similar results also hold for cartesian product of graphs i.e., if G, H ∈ Ω then G□H ∈ Ω . Notation 2.1. For an even order simple connected graph G and a positive integer d, we say that G ∈ Ψd if there exists a partition of V (G) into two sets U1 = {u i : 0 ⩽ i ⩽ n2 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ n2 − 1} so that the following conditions are satisfied ′ 1. d(u i , u i+1 ) ⩾ d, d(u i′ , u i+1 )⩾d ′ 2. d(u i+1 , u i ) ⩾ d + 1 ′ 3. d(u i , u i+1 ) ⩾ d − 1.

Notation 2.2. For an odd order simple connected graph G and a positive integer d, we say that G ∈ Ψd′ if there exist a vertex u and a partition of V (G) − {u} into two sets, U1 = {u i : 0 ⩽ i ⩽ n−1 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ n−1 − 1} 2 2 so that the following conditions are satisfied ′ 1. d(u i , u i+1 ) ⩾ d, d(u i′ , u i+1 )⩾d ′ 2. d(u i+1 , u i ) ⩾ d + 1

Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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′ 3. d(u i , u i+1 )⩾d −1 4. d(u n−1 −1 , u) = d or d(u 0 , u) = d 2

5. d(u ′n−1

2 −1

, u) = d or d(u 1 , u) = d − 1.

In the theorem below we give conditions for which the upper bounds of Theorems 2.1 or 2.2 and the lower bound of Theorem 2.3 coincide. Theorem 2.5. Let G be an n-vertex with diameter diam(G) and triameter M. Suppose d is ⌊ simple⌋ connected graph ) ( a positive integer such that d ⩽ diam(G)+1 . If G ∈ Ω ∩ Ψ ∪ Ψd′ and M = diam(G)+1+2d or diam(G)+2+2d, d 2 then ⎧ ⎪ (diam(G) + 1 − d) (n − 2) ⎪ ⎨ + 1, if n is an even integer; 2 r n(G) = ⎪ ⎪ ⎩ (diam(G) + 1 − d) (n − 1) , if n is an odd integer. 2 Proof. The result follows immediately from Theorems 2.1, 2.2, and 2.3. □ 3. Upper bound for other r ck (G) The following theorem gives an upper bound for r ck (G), 1 ⩽ k < diam(G), of an arbitrary graph G of even order. Theorem 3.1. Let G be a simple connected graph with 2m vertices and diameter diam(G). For a partition of the vertex set V (G) into two sets U1 = {u i : 0 ⩽ i ⩽ m − 1} and U2 = {u i′ : 0 ⩽ i ⩽ m − 1}, if di and l j for 0 ⩽ i ⩽ m − 2 and 0 ⩽ j ⩽ m − 1 are non-negative integers such that ′ ′ (i) d(u i , u i+1 ), d(u i′ , u i+1 ), d(u i , u i+1 ) and d(u i+1 , u i′ ) are greater than or equal to di for 0 ⩽ i ⩽ m − 2; (ii) d(u i , u i′ ) ⩾ li for 0 ⩽ i ⩽ m − 1;

then for a positive integer k with 1 ⩽ k ⩽ diam(G) − 1, r ck (G) ⩽

m−2 ∑ i=0

ai +

m−1 ∑

max{k + 1 − li , 0},

i=0

where a0 , a1 , . . . , am−2 are given by ⎧ k +⌉1 − di , if 2di ⩽ k + 1; ⎨⌈ k ai = , if 2di ⩾ k + 2. ⎩ 2 Moreover, if all di and li are greater than k then ⎧( ) ⎪ m−2 ⎪ ⎪ k, if m ≡ 0 (mod 2), ⎨ r ck (G) ⩽ ( 2 ) ⎪ m−1 ⎪ ⎪ ⎩ k, if m ≡ 1 (mod 2). 2 ⌈ ⌉ ⌈ ⌉ ⌊ ⌋ Proof. If 2di ⩾ k + 2 then ai = k2 and k + 1 − di ⩽ k + 1 − k+2 = k2 . So ai ⩾ max{k + 1 − di , 0} for 2 0 ⩽ i ⩽ m − 2. Next we show that ai + ai+1 ⩾ k for 0 ⩽ i ⩽ m − 3. If both 2di and 2di+1 are greater than k + 2 then the inequality is obviously true. If both 2di , 2di+1 ⩽ k + 1 then ai + ai+1 = 2(k +⌊1) −⌋di − ⌈ kd⌉i+1 ⩾ k + 1. Finally, if, either 2di or 2di+1 is greater than or equal to k + 2 then ai + ai+1 ⩾ k + 1 − k+1 + ⩾ k. Let us 2 2 Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

8

L. Saha / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx Table 3 Comparison between the values of max{k + 1 − d(u, v), 0} and | f (u) − f (v)|. Pair of vertices u, v ∈ U1 ∪ U2

j −i

d(u, v)

max{k + 1 − d(u, v)}

| f (u) − f (v)|

u, v ∈ U1 , or u, v ∈ U2

1 ⩾2

⩾ di ⩾1

⩽ max{k + 1 − di , 0} ⩽k

⩾ ai ⩾ max{k + 1 − di , 0} ⩾ ai + ai+1 ⩾ k

u ∈ U1 , v ∈ U2 or u ∈ U2 , v ∈ U1

0 1 ⩾2

li ⩾ di ⩾1

max{k + 1 − li , 0} ⩽ max{k + 1 − di , 0} ⩽k

max{k + 1 − li , 0} ⩾ ai − 1 ⩾ max{k + 1 − di , 0} ⩾ ai + ai+1 ⩾ k

take a−1 = 0 and l−1 = k + 1. Next we define a coloring f of G as follows: 0 ⩽ i ⩽ m − 1, f (u i ) =

i ∑

(a j−1 + max{k + 1 − l j−1 , 0})

j=0

f (u i′ ) = f (u i ) + max{k + 1 − li , 0}. In Table 3, we check the radio condition for f . Without loss of generality, let i ⩽ j and for u, v ∈ U1 ∪ U2 we take u ∈ {u i , u i′ } and v ∈ {u j , u ′j }. Since ai ⩾ max{k + 1 − di , 0} for 0 ⩽ i ⩽ m − 2 and ai + ai+1 ⩾ k for 0 ⩽ i ⩽ m − 3, we obtain Table 3. From Table 3, it is clear that | f (u) − f (v)| ⩾ max{k + 1 − d(u, v), 0} for all ∑m−2 ∑m−1 u, v ∈ V . Therefore f is a radio k-coloring of G and the span of f is i=0 ai + i=0 max{k + 1 − li , 0}. Now we take all di and li greater than k. Let m = r (mod 2) where r = 0, 1. We define a coloring f as ′ ) = f (u i′ ), for i ∈ {0, 2, . . . , m − r − 2}. If r = 1 then we give the f (u i ) = ik2 = f (u i′ ), f (u i+1 ) = f (u i ) , f (u i+1 = f (u ′m−1 ). Then clearly, f is a radio k-coloring of G color to the vertices u m−1 and u ′m−1 as f (u m−1 ) = (m−1)k 2 (m−2)k (m−1)k with span 2 or 2 according to m is an even or an odd, respectively. □ In the theorem below we give an upper bound for r ck (G), 1 ⩽ k < diam(G), of an arbitrary graph of odd order. Theorem 3.2. Let G be a graph with 2m + 1 vertices and diameter diam(G). For a partition of the vertex set V (G) − {u}, where u is a vertex of G, into two sets U1 = {u i : 0 ⩽ i ⩽ m − 1} and U2 = {u i′ : 0 ⩽ i ⩽ m − 1}, if di and li , 0 ⩽ i ⩽ m − 1, are non-negative integers such that ′ ′ (a) d(u i , u i+1 ), d(u i′ , u i+1 ), d(u i , u i+1 ) and d(u i+1 , u i′ ) are greater than or equal to di for 0 ⩽ i ⩽ m − 2; ′ (b) dm−1 = min{d(u, u 0 ), d(u, u 0 )} or min{d(u, u ′m−1 ), d(u, u m−1 )} according to d(u, u 0 ) ⩾ d(u, u ′m−1 ) or d(u, u 0 ) < d(u, u ′m−1 ), r espectively (c) d(u i , u i′ ) ⩾ li for 0 ⩽ i ⩽ m − 1;

then for a positive integer k with 1 ⩽ k ⩽ diam(G)−1, r ck (G) ⩽

m−1 ∑

(ai +max{k +1−li , 0}), where a0 , a1 , . . . , am−1

i=0

are given by { k + 1 − di , if 2di ⩽ k + 1, k ai = ⌈ ⌉, if 2di ⩾ k + 2. 2 Moreover, if all di and li are greater than k then ⎧ ⎪ mk ⎪ ⎪ ⎨ , if m ≡ 0 (mod 2), r ck (G) ⩽ (2 ) ⎪ ⎪ m−1 ⎪ k, if m ≡ 1 (mod 2). ⎩ 2 Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

L. Saha / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx

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Proof. We give the proof of this theorem by considering the following two cases. Case I: In this case we take dm−1 = min{d(u, u 0 ), d(u, u ′0 )}. Then we get d(u, u 0 ) ⩾ dm−1 , d(u, u ′0 ) ⩾ dm−1 . Let us take a−1 = am−1 , l−1 = k + 1 and we define a coloring f of G as follows. For 0 ⩽ i ⩽ m − 1, f (u) = 0, i ∑ ( ) f (u i ) = a j−1 + max{k + 1 − l j−1 , 0} , j=0

f (u i′ )

= f (u i ) + max{k + 1 − li , 0}.

If v, w ∈ V − {u} then by the similar way as in Theorem 3.1 we can show that | f (v) − f (w)| ⩾ max{k + 1 − d(v, w), 0}. Now, let w ∈ V − {u}. If w = u 0 , then f (w) − f (u) = a−1 + max{k + 1 −l−1 , 0} = am−1 ⩾ k + 1 − dm−1 and if w = u ′0 then f (w)− f (u) ⩾ am−1 ⩾ k+1−dm−1 . From the definition of f it is clear that if w ∈ V −{u, u 0 , u ′0 }, then | f (u) − f (w)| ⩾ k. So | f (v) − f (w)| ⩾ max{k + 1 − d(v, w), 0} for all v, w ∈ V i.e., f is a radio k-coloring of G. Case II: Here we take dm−1 = min{d(u, u ′m−1 ), d(u, u m−1 )}. Then we get, d(u, u m−1 ) ⩾ dm−1 and d(u, u ′m−1 ) ⩾ dm−1 . In this case we take a−1 = 0, l−1 = k + 1 and define a coloring f of G as follows. For 0 ⩽ i ⩽ m − 1, f (u i ) =

i ∑ (a j−1 + max{k + 1 − l j−1 , 0}), j=0

f (u i′ ) = f (u i ) + max{k + 1 − li , 0} f (u) = f (u ′m−1 ) + am−1 . Similar to Case I, we can show that f is a radio k-coloring of G and the span of f is

m−1 ∑

(ai +max{k +1−li , 0}).

□

i=0

4. Application of bounds to some classes of graphs In this section we apply the bounds proved in previous two sections to the graphs Cn and Q n . 4.1. Upper bound for r ck (Cn ) In this subsection we give an upper bound for r ck (Cn ) when n ≡ 0, 1, 2, 3 (mod 4). For this, we need the following lemmas. Lemma 4.1. If n ≡ 0 (mod 4), then the vertex set {v0 , v1 , . . . , vn−1 } of Cn has a partition into two sets U1 = {u i : 0 ⩽ i ⩽ n2 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ n2 − 1} such that (i) d(u i , u i′ ) = n2 = diam(Cn ) for 0 ⩽ i ⩽ n2 − 1; ′ ) for all i ∈ {0, 2, . . . , n2 − 2} and d(u i , u i+1 ) = (ii) d(u i , u i+1 ) = n4 = d(u i′ , u i+1 n i ∈ {1, 3, . . . , 2 − 3}; ′ (iii) d(u i+1 , u i′ ) = n4 = d(u i , u i+1 ) for all i ∈ {0, 2, . . . , n2 − 2} and d(u i+1 , u i′ ) = n i ∈ {1, 3, . . . , 2 − 3}.

n 4

′ − 1 = d(u i′ , u i+1 ) for all

n 4

′ + 1 = d(u i , u i+1 ) for all

Proof. Let τ, τ ′ : {0, 1, . . . , n2 − 1} → {0, 1, . . . , n − 1} be two functions defined by ⎧ i ⎪ ⎨ , if i ≡ 0 (mod 2); τ (i) = 2n i − 1 ⎪ ⎩ + , if i ≡ 1 (mod 2); 2 ⎧ n4 i ⎪ if i ≡ 0 (mod 2); ⎨ + , τ ′ (i) = 2 2 ⎪ ⎩ 3n + i − 1 , if i ≡ 1 (mod 2). 4 2 Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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L. Saha / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx

Let u i = vτ (i) , u i′ = vτ ′ (i) for 0 ⩽ i ⩽ n2 − 1, U1 = {u i : 0 ⩽ i ⩽ n2 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ U1 ∪ U2 is a partition of V (Cn ) which satisfies the conditions (i), (ii) and (iii) of the lemma. □

n 2

− 1}. Then

Lemma 4.2. If n ≡ 2 (mod 4), then the vertex set {v0 , v1 , . . . , vn−1 } of Cn has a partition into two sets U1 = {u i : 0 ⩽ i ⩽ n2 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ n2 − 1} such that (i) d(u i , u i′ ) = n2 = diam(Cn ) for 0 ⩽ i ⩽ n2 − 1; ′ ) for 0 ⩽ i ⩽ n2 − 2; (ii) d(u i , u i+1 ) = n−2 = d(u i′ , u i+1 4 n+2 ′ (iii) d(u i+1 , u i′ ) = 4 = d(u i , u i+1 ) for 0 ⩽ i ⩽ n2 − 2. Proof. Let n = 4 p + 2 and τ, τ ′ : {0, 1, . . . , n2 − 1} → {0, 1, . . . , n − 1} be two functions defined by τ (i) = (i(3 p + 2))

(mod n), τ ′ (i) = (i(3 p + 2) + 2 p + 1)

(mod n), 0 ⩽ i ⩽ 2 p.

Suppose u i = vτ (i) , = vτ ′ (i) , 0 ⩽ i ⩽ − 1, U1 = {u i : 0 ⩽ i ⩽ n2 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ U1 ∪ U2 is a partition of V (Cn ) which satisfies the conditions (i), (ii) and (iii) of the lemma. □ u i′

n 2

n 2

− 1}. Then

Lemma 4.3. If n ≡ 1 (mod 4), then the vertex set {v0 , v1 , . . . , vn−1 } of Cn has a partition into two sets U1 = {u i : 0 ⩽ i ⩽ n−1 } and U2 = {u i′ : 0 ⩽ i ⩽ n−1 − 1} such that 2 2 (i) (ii) (iii) (iv) (v)

d(u i , u i′ ) = n−1 = diam(Cn ) for 0 ⩽ i ⩽ n−1 − 1; 2 2 n−1 n−1 n−5 for 0 ⩽ i ⩽ 2 − 1, d(u i , u i+1 ) = 4 or 4 according ′ for 0 ⩽ i ⩽ n−1 − 2, d(u i′ , u i+1 ) = n−1 or n−5 according 2 4 4 n−1 n−1 ′ for 0 ⩽ i ⩽ 2 − 2, d(u i+1 , u i ) = 4 or n+5 according 4 n−1 n−5 ′ for 0 ⩽ i ⩽ n−1 − 2, d(u , u ) = or according i i+1 2 4 4

to to to to

i i i i

is is is is

even even even even

or or or or

odd, odd, odd, odd,

respectively; respectively; respectively; respectively.

Proof. We define a mapping τ on {0, 1, . . . , n − 1} as ⎧ i ⎪ ⎨ , if i ≡ 0 (mod 2); τ (2i) = n2 − 1 i − 1 ⎪ ⎩ + , if i ≡ 1 (mod 2); 2 ⎧ 4 n−1 ⎪ ⎨τ (2i) + , if i ≡ 0 (mod 2); 2 τ (2i + 1) = n+1 ⎪ ⎩τ (2i) + , if i ≡ 1 (mod 2); 2 ⎧ i n−1 ⎪ ⎨ + , if i ≡ 0 (mod 2); 2 = i −1 2n+1 ⎪ ⎩ + , if i ≡ 1 (mod 2); 2 2 3(n − 1) τ (n − 1) = . 4 Suppose u i = vτ (2i) for 0 ⩽ i ⩽ n−1 , u i′ = vτ (2i+1) for 0 ⩽ i ⩽ n−1 − 1, U1 = {u i : 0 ⩽ i ⩽ n−1 } and 2 2 2 n−1 ′ U2 = {u i : 0 ⩽ i ⩽ 2 − 1}. Then U1 ∪ U2 is a partition of V (Cn ) which satisfies the conditions (i), (ii), (iii), (iv) and (v) of the lemma. □ Lemma 4.4. Let n ≡ 3 (mod 4), and V (Cn ) = {v0 , v1 , . . . , vn−1 } be the vertex set of Cn . Then V (Cn ) \ {u}, for some vertex u, has a partition into two sets U1 = {u i : 0 ⩽ i ⩽ n−1 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ n−1 − 1} which 2 2 satisfies the conditions (i) (ii) (iii) (iv)

d(u i , u i′ ) = n−1 = diam(Cn ) for 0 ⩽ i ⩽ n−1 − 1; 2 2 n−1 n+1 for 0 ⩽ i ⩽ 2 − 2, d(u i , u i+1 ) = n−3 or according as i is an even or an odd integer; 4 4 n−3 n+1 ′ ′ for 0 ⩽ i ⩽ n−1 − 2, d(u , u ) = or according as i is an even or an odd integer; i i+1 2 4 4 n−1 d(u i+1 , u i′ ) ⩾ n+1 for 0 ⩽ i ⩽ − 1; 4 2

Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

L. Saha / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx ′ (v) d(u i , u i+1 ) ⩾ n−3 for 0 ⩽ i ⩽ 4 (vi) d(u 0 , u) = d(u ′0 , u) = n+1 . 4

n−1 2

11

− 2;

Proof. For n = 4 p + 3, we define τ on {0, 1, . . . , n − 1} as, τ (4i) τ (4i + 1) τ (4i + 2) τ (4i + 3) τ (4 p + 2)

= = = = =

4 p + 2 + i (mod n), 0 ⩽ i ⩽ p; 2 p + 1 + i (mod n), 0 ⩽ i ⩽ p; 3 p + 2 + i (mod n), 0 ⩽ i ⩽ p − 1; p + 1 + i (mod n), 0 ⩽ i ⩽ p − 1; p.

Suppose u i = vτ (2i) and u i′ = vτ (2i+1) for 0 ⩽ i ⩽ n−1 − 1; u = v p ; U1 = {u i : 0 ⩽ i ⩽ n−1 } and 2 2 n−1 ′ U2 = {u i : 0 ⩽ i ⩽ 2 − 1}. Then U1 ∪ U2 is a partition of V (Cn ) \ {u} which satisfies the conditions (i) − (vi). □ Theorem 4.1. For a cycle Cn , ⎧( )( ) ⎪ n+8 n−2 ⎪ ⎪ + 1, ⎪ ⎪ ⎪ ⎪( 4 ) ( 2 ) ⎪ ⎪ ⎪ n−2 ⎪ n+7 ⎪ + 1, ⎨ r n(Cn ) ⩽ ( 4 ) ( 2 ) ⎪ n−2 n+6 ⎪ ⎪ ⎪ + 1, ⎪ ⎪ 4 2 ⎪ ⎪ ( ) ( ) ⎪ ⎪ ⎪ n+5 n−1 ⎪ ⎩ , 4 2

if n ≡ 0

(mod 4);

if n ≡ 1

(mod 4);

if n ≡ 2

(mod 4);

if n ≡ 3

(mod 4).

Proof. For even integer n we consider the partition {U1 , U2 } as in the proof of Lemmas 4.1 and 4.2. We take or n−2 , 0 ⩽ i ⩽ n2 − 2, according as n ≡ 0 (mod 4) or n ≡ 2 (mod 4) and li = n2 for 0 ⩽ i ⩽ n2 − 1. Then di = n−4 4 4 the partition {U1 , U2 } satisfies all the conditions of Theorem 2.1 with these di ’s and li ’s. Thus from Theorem 2.1, we have n

r n(Cn ) ⩽

2 −2 ( ∑ n

+ 1 − di

)

2 i=0 ⎧( n + 8 ) ( n − 2 ) ⎪ ⎪ + 1, if n ≡ 0 ⎨ 4 2 = ( )( ) ⎪ n+6 n−2 ⎪ ⎩ + 1, if n ≡ 2 4 2

(mod 4); (mod 4);

For n ≡ 3 (mod 4), we consider the partition {U1 , U2 } as in the proof of Lemma 4.4. We take di = n−3 and 4 n−1 li = n−1 , 0 ⩽ i ⩽ − 1, and verify that {U , U } satisfies all the conditions of Theorem 2.2 (to use Theorem 2.2 1 2 2 2 we consider u is the last vertex of U1 ) with these di ’s and li ’s. Thus from Theorem 2.2, we have n−1 −1 2

) ∑ (n − 1 n−3 r n(Cn ) ⩽ +1− 2 4 i=0 ( )( ) n+5 n−1 = . 4 2 For n ≡ 1 (mod 4) similar logic may be used to prove the result. □ Remark 4.1. We can show that this upper bound of r n(Cn ) coincides with the lower bound of Theorem 2.3 when n ≡ 0, 2, 3 (mod 4). Hence the upper bound obtained from Theorems 2.1 and 2.2 is sharp for radio number of cycle Cn when n ≡ 0, 2, 3 (mod 4) (which was settled in [5] by a different approach). Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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L. Saha / AKCE International Journal of Graphs and Combinatorics xxx (xxxx) xxx

For n ≡ 1, 2, 3 (mod 4) the antipodal number of Cn was determined by Jaun et al. [16]. Also they have given an upper and a lower bound for the same when n ≡ 0 (mod 4). In the theorem below we give upper bounds for r ck (Cn ) with k < diam(Cn ) − 1. Theorem 4.2. For a cycle Cn of length n, we have the following: (a) For n ≡ 0 (mod 4), ⎧ ⎪ n(n − 4) ⎪ ⎪ + 1, if k = n2 − 2; ⎪ ⎪ ⎪ 8 ⎪ ) ⎨⌈ ⌉ ( n−2 k r ck (Cn ) ⩽ , if n4 ⩽ k ⩽ n2 − 3; ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ k(n − 4) , if 1 ⩽ k ⩽ n−4 . 4 4 (b) For n ≡ 2 (mod 4), ⎧ n −2 ( ) 2 ⎪ ∑ ⎪ n−2 ⎪ ⎪ k+1− , if k = n2 − 2; ⎪ ⎪ 4 ⎪ ⎪ ⎨⌈i=0 ⌉ ( ) k n−2 r ck (Cn ) ⩽ ⩽ k ⩽ n−6 ; , if n−2 ⎪ ⎪ 4 2 ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ k(n − 4) ⎩ , if 1 ⩽ k ⩽ n−2 − 1. 4 4 (c) For n ≡ 1 (mod 4), ⎧ n−1 −1 ( ) ⎪ 2 ⎪ ∑ ⎪ n−5 ⎪ ⎪ k + 1 − , if n−7 ⩽ k ⩽ n−5 ; ⎪ 2 2 ⎪ ⎪ 4 ⎪ i=0 ⎨⌈ ⌉ ( ) r ck (Cn ) ⩽ k n−2 ⎪ ⩽ k ⩽ n−7 − 1; , if n−2 ⎪ 3 2 ⎪ 2 2 ⎪ ⎪ ⎪ ⎪ ⎪ k(n − 4) ⎪ ⎩ , otherwise. 4 (d) For n ≡ 3 (mod 4), ⎧ n−1 −1 ( ) ⎪ 2 ⎪ ∑ ⎪ n−3 ⎪ ⎪ − 2; k + 1 − , if k = n−1 ⎪ 2 ⎪ ⎪ 4 ⎪ ⎨⌈i=0⌉ ( ) r ck (Cn ) ⩽ k n−2 ⎪ , if n−3 ⩽ k ⩽ n−1 − 3; ⎪ 2 4 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ k(n − 4) ⎪ ⎩ , if 1 ⩽ k ⩽ n−3 − 1. 4 4 ⌈ ⌉ Proof. First we take n is an even integer. Let di = n−4 , 0 ⩽ i ⩽ n2 − 2 and li = n2 , 0 ⩽ i ⩽ n2 − 1. Using 2 Theorem we get the desired result for even integer n. For n ≡ 3 (mod 4), we take ⌊ 3.1, ⌉ Lemmasn 4.1 and 4.2, n−1 di = n−4 and l = , 0 ⩽ i ⩽ − 1. With these di ’s and li ’s, from Theorem 3.2 and Lemma 4.4, we get the i 2 2 2 desired result for n ≡ 3 (mod 4). For n ≡ 1 (mod 4), using Theorem 3.2 and Lemma 4.3 we can also prove the result. □ 4.2. Upper bound for r ck (Q n ) In this subsection we give an upper bound for r ck (Q n ) which is an improvement of that given by Kchikech et al. [17]. The following definitions are given by Khennoufa et al. [6]. Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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Definition 4.1. For two positive integers n and l with l < n, a binary (n, l)-Gray code is a sequence of all n-bit binary strings such that the Hamming distance between any two successive strings is exactly l (where Hamming distance between two strings is the number of places where they differ). Definition 4.2. For two positive integers n and l with l < n, a quasi (n, l)-Gray code is a sequence of all n-bit binary strings such that the Hamming distance between any two successive strings is exactly l except between the (2n−1 − 1)-th term and 2n−1 -th terms for which it is l − 1 or l + 1. Notation 4.1. For any positive integer n, we define δ(n) as { 0, if n ≡ 2 (mod 4), δ(n) = 1, if n ̸≡ 2 (mod 4). The following theorem gives a partition of the vertex set of Q n satisfying some conditions. Theorem 4.3. For a hypercube Q n of dimension n, there exists a partition of the vertex set of Q n into two sets U1 = {u i : 0 ⩽ i ⩽ 2n−1 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ 2n−1 − 1} satisfying the following properties ⌊ ⌋ ′ (i) d(u i , u i+1 ) = n2 = d(u i′ , u i+1 ) for all i = 0, 1, . . . , 2n−1 − 2 except i = 2n−2 − 1; ⌊n⌋ ⌊n⌋ (ii) d(u i , u i+1 ) = 2 − δ(n) or 2 + δ(n) for i = 2n−2 − 1; ⌊ ⌋ ⌊ ⌋ ′ (iii) d(u i′ , u i+1 ) = n2 − δ(n) or n2 + δ(n) for i = 2n−2 − 1; (iv) d(u i , u i′ ) = n for all i = 0, 1, . . . , 2n−1 − 1; (v) d(u i+1 , u i′ ) = n − d(u i , u i+1 ) for all i = 0, 1, . . . , 2n−1 − 2; ′ ′ (vi) d(u i , u i+1 ) = n − d(u i′ , u i+1 ) for all i = 0, 1, . . . , 2n−1 − 2; where δ(n) is defined as in Notation 4.1. Proof. Let Q n−1 and Q ′n−1 be two copies of (n − 1)-dimensional hypercubes in Q n induced by the strings with a 0 and an 1 at the beginning, respectively. Let u 0 , u 1 , . . . , u 2n−1 −1 be the ordering of the vertices of Q n−1 induced by a quasi (n − 1, ⌊ n2 ⌋)-Gray code if n ̸≡ 2 (mod 4) and (n − 1, n2 )-Gray code if n ≡ 2 (mod 4) (such code exists, see [6]). If we define u i′ = u¯ i (where u¯ i is obtained from u i by flipping 0’s and 1’s), 0 < i ⩽ 2n−1 − 1, then u ′0 , u ′1 , . . . , u ′2n−1 −1 is an ordering of the vertices Q ′n−1 induced by the same Gray code as in above. By taking U1 = {u i : 0 ⩽ i ⩽ 2n−1 − 1} and U2 = {u i′ : 0 ⩽ i ⩽ 2n−1 − 1}, we get the desired result. □ The following upper bound for r ck (Q n ) was determined by Kchikech et al. [17]. However, this bound is quite far from being optimal, specially when k is close to the diameter of Q n . Theorem 4.4. For the hypercube Q n of dimension n ⩾ 2 and for any positive integer k ⩾ 2, r ck (Q n ) ⩽ (2n − 1)k − 2n−1 + 1. For 1 ⩽ k ⩽ n − 1, the theorem below gives an improvement of the above upper bound for r ck (Q n ). Theorem 4.5. For the hypercube Q n of dimension n ⌊n ⌋ ⎧ n−1 ⎪ (2 − 1)(k + 1 − ) + δ(n), ⎪ ⎪ 2 ⎪ ⌈ ⌉ ⎨ k + δ(n), r ck (Q n ) ⩽ k2n−2 − ⎪ 2 ⎪ ⎪ ⎪ ⎩k2n−2 − k,

⩾ 2 and a positive integer k with 2 ⩽ k ⩽ n − 1, ⌊n ⌋ if 2 − 2 ⩽ k ⩽ n − 1; 2 ⌊n ⌋ ⌊n ⌋ if −1⩽k ⩽2 − 3; 2 2 ⌊n ⌋ if 2 ⩽ k ⩽ − 2; 2

where δ(n) is defined as in Notation 4.1. Proof. To prove this theorem we take the same partition of V (Q n ) as in Theorem 4.3. Then by Theorem 3.1 we get the desired result. □ Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.

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It is easy to verify that for even integer n the bound given in Theorem 4.5 includes the antipodal number r cn−1 (Q n ) (which was settled in [6] by a different approach). The theorem below gives the value of nearly antipodal number of Q n which is obtained from Theorems 4.5 and 2.4 by taking k = n − 2. Theorem 4.6. For the hypercube Q n of dimension n, we have the following ( ) (i) if n ≡ 2 (mod 4), then r cn−2 (Q n ) = (2n−1 − 1) n−2 ; 2 (ii) if n ≡ 0 (mod 4), then ) ( ) ( ( ) n−2 n−2 ⩽ r cn−2 (Q n ) ⩽ 2n−1 − 1 + 1. (2n−1 − 1) 2 2 5. Concluding remark This article presented a method of getting an upper bound on r ck (G) for general graphs G, by finding a radio k-coloring to a specific partition of the vertex set of G into two sets with certain conditions. Then we apply the results to cycles and hypercubes. Acknowledgments The author is thankful to the referees for their valuable comments and suggestions which improved the presentation of the paper. Also the author is thankful to the National Board for Higher Mathematics (NBHM), India for its financial support (Grant Nos. 2/48(22)/R & D II/4033). References [1] W.K. Hale, Frequency assignment, Theory and application, Proc. IEEE. 68 (1980) 1497–1514. [2] G. Chartrand, D. Erwin, P. Zhang, A graph labeling problem suggested by FM channel restrictions, Bull. Inst. Combin. Appl. 43 (2005) 43–57. [3] G. Chartrand, D. Erwin, F. Harrary, P. Zhang, Radio labeling of graphs, Bull. Inst. Combin. Appl. 33 (2001) 77–85. [4] P. Zhang, Radio labelings of cycles, Ars Combin. 65 (2002) 21–32. [5] D. Liu, X. Zhu, Multi-level distance labelings for paths and cycles, SIAM J. Discrete Math. 19 (3) (2005) 610–621. [6] R. Khennoufa, O. Togni, The radio antipodal and radio numbers of the hypercube, Ars Combin. 102 (2011) 447–461. [7] S.R. Kola, P. Panigrahi, An improved lower bound for the radio k-chromatic number of the Hypercube Q n , Comput. Math. Appl. 60 (7) (2010) 2131–2140. [8] J.P. Ortiz, P. Martinez, M. Tomova, C. Wyels, Radio numbers of some generalized prism graphs, Discuss. Math. Graph Theory. 31 (1) (2011) 45–62. [9] X. Li, V. Mak, S. Zhou, Optimal radio labellings of complete m-ary trees, Discrete Appl. Math. 158 (2010) 507–515. [10] M. Morris-Rivera, M. Tomova, C. Wyels, Y. Yeager, The radio number of Cn □Cn , Ars Combin. 103 (2012) 81–96. [11] L. Saha, P. Panigrahi, On the radio number of toroidal grids, Australian J. Combin. 55 (2013) 273–288. [12] D.D.-F. Liu, M. Xie, Radio number for square paths, Ars Combin. 90 (2009) 307–319. [13] D.D.-F. Liu, M. Xie, Radio number for square of cycles, Congr. Numer. 169 (2004) 105–125. [14] L. Saha, P. Panigrahi, A lower bound for radio k-chromatic number, Discrete Appl. Math. 192 (2015) 87–100. [15] S. Das, S.C. Ghosh, S. Nandi, S. Sen, A lower bound technique for radio k-coloring, Discrete Math. 340 (5) (2017) 855–861. [16] J. Jaun, D.D.-F. Liu, Antipodal labelings of cycles, Ars Combin. 103 (2012) 81–86. [17] M. Kchikech, R. Khennoufa, O. Togni, Radio k-labelings for cartesian products of graphs, Electron. Notes Discrete Math. 22 (2005) 347–352.

Please cite this article as: L. Saha, Upper bound for radio k-chromatic number of graphs in connection with partition of vertex set, AKCE International Journal of Graphs and Combinatorics (2019), https://doi.org/10.1016/j.akcej.2019.03.024.