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257
AMS Subject Classification: 05C78
SOME RESULTS ON PRIME LABELINGS OF GRAPHS
ADEL T. DIAB
Ain Shams University e-mail: [email protected]
A graph is said to be prime if it has a relatively prime labeling on its vertices which satisfies certain properties. The purpose of this paper is to give some new families of graphs that have a prime labeling and give some necessary and sufficient conditions for some families of prime graphs.
Key words: Graph, prime labeling.
Introduction
It is well known that graph theory has applications in many other fields of study, including physics, chemistry, biology, communication, psychology, sociology, economics, engineering, operations research, and especially computer science. For all standard notation and terminology in graph theory we follow [10]. Graph labelings where the vertices are assigned real values subject to certain conditions like as Graceful [9,12], Harmonious [9,12], Cordial [3-8], Prime [9] and others, have often been motivated by practical problems such as coding theory, communication networks and astronomy, but they are also of logico- mathematical interest in their own right. An enormous body of literature has grown around the subject especially in the last thirty years or so, and is presented in a survey by Gallian [9].
A graph G is an ordered pair G(V,E), where V(G) stands for a finite set of elements called vertices, while E(G) - a finite set of unordered pairs of vertices called edges. The cardinality of the set of vertices V(G) is denoted by the symbol V’| and called the order of graph G. Likewise, the cardinality of the set of edges E(G) is denoted by the symbol \E and called the size of graph G. The vertices u, v E V(G) are called adjacent (or neighbors) if {ii.i7} in E(G) and nonadjacent if not in E(G). The degree deg(v) of vertex v in graph G is the number of edges incident to vertex v in graph G, i.e., |e E E (G): v E e |. The maximum degree of a vertex in graph G is denoted by A(G), while the minimum degree is denoted by 8{G). A set of vertices V of a graph G is said to be independent if any two vertices u and v in V are not adjacent in G.
A graph G with vertex set V(G) is said to have a prime labeling if its vertices can be labeled with distinct integers 1, 2, 3, ..., |V| such that for each xy E £(G) the labels assigned to x and y are relatively prime. A graph that admits a prime labeling is called a prime graph. The notion of a prime labeling originated with Entringer and was introduced in a paper by Tout, Dabboucy, and Howalla [14]. Around the classes of trees known to have a prime labelings are: paths, stars, caterpillars , complete binary trees, spiders and all tree of order up to 50. Also, other graphs with prime labelings include all cycles and a complete graph Kn does not have a prime labeling for n > 4 [9].
Youssef [15] has shown that O K-l is prime if and only if n < 7. In section 3, we extend this result to show that Kn 0 K2 is prime if and only if n S 16 . Moreover, we show that the graph Km 0 Kn is prime if and only if m = 1 and n = 2, or n = 1 and for all m > 1.
Deretsky, Lee and Mitchem [2] have shown that the disjoint union C2!i U Cn of two cycles C2k and Cn is prime. In section 4, we extend this result to show that the union Cn U Cm of two cycles Cn and Cm is not prime if and only if both n and m are odd. Moreover, we show that the joint Cn 4- Cm of two cycles Cn and Cm is not prime for all n > 3 and m > 3, the union Pn U Pm of two paths and Pm is prime for all n > 1 and m > I, the joint Pn 4- Pm of two paths Pn and Pm is prime if and only if n =1 and m > l(or vice versa ), or n =2 and m odd (or vice versa ), the union U Pm of cycles Cn and paths PW! is prime for all ft > 3 and m > 1, and the joint Cn 4 Pm of cycles Cn and paths Pm is not prime for all n > 3 and m > 1.
Seoud, Diab, and Elsakhawi [12], have shown the following complete bipartite graphs are prime: K2 m and K3 m unless m = 3 or m = 7. In section 5, we extend those results to some complete tripartite, namely, is prime for all n > 1 and K12in is prime for all n > 1 except n = 3 or n = 7. Finally, we show that the following graphs are prime: Pn U Kl m for all n > 1 and m > 1, Cn U Kl m for all n > 3 and m > 1, Kn U Kl m for all n > 1 and m > 1, 4- Kl m for all m > 1, Pn O for all n > 1, O Pn for
all 7i > 1, and Kn U Km for all n > I and m '■> I.
2. Notation and Preliminaries
We introduce some basic properties of graphs and we will study the effect of these properties on the prime graphs. The maximum cardinality of an independent set of vertices of a graph G is called the vertex independence number and is denoted by (3(G).
A coloring of a graph G is an assignment of colors (which are actually considered as elements of some set) to the vertices of G, one color to each vertex, so that adjacent vertices are assigned different colors. A graph G for which there exists a vertex - coloring which requires k colors is called k- colorable, while such a coloring is called a k- coloring. The smallest number k for which there exists a k-coloring of graph G is called the chromatic number of graph G and is denoted by j(G). Such a graph G is called k-chromatic, while any coloring of G which requires k = /(C) colors is called chromatic or optimal. Harary [10] has shown that for any graph G, we have j(G) < \V\ — 1 —
The clique number oi(G) of a graph G is the maximum order among the complete subgraph of G. Clearly, co(C) = P(C) for every graph G, where the complement G of a graph G is that graph with vertex set V(G) such that two vertices are adjacent in G if and only if these vertices are not adjacent in G. If Kn Q G for some n, where Kn is a complete graph of order n, then x(G) > n . It follows that jU7) > &>(£) ( for more details, one can refer to [1,2,9,10]).
We follow the basic notation and terminology of the theory of numbers as in [11]. In particular, we let pr be the rth prime number, where px = 2 ; n(n) is the number of primes less than or equal to n ; the Euler's <t> - function cf>(n) is defined as the number of positive integers less than or equal to n that are relatively prime to n.
Let x be any real number, then x and.t denote the greatest integer less than or equal x, the smallest integer greater than or equal x respectively.
Seoud and Youssef [13,15] proved the following result, which gives some necessary conditions for a prime graph.
Theorem 2.1. If G is a prime graph of order n, then
(1) |£ j < p(n). where p(n) is the maximum number of edges in G,
■ ..
Proof. For the proof, the reader may be referred to the articles [13,15].
Finally, for specific labelings of Kn 0 Km , we let V(A„ 0 A",,;) = { :
1 < i < n} (J { Vjj : 1 < j < m }, and we write the vertices of Kn 0 Km in m- tuples
(uh vu , v2i, labeling.
,vmi ), where 1 < i < n and we let ( xh yu , y2, ) denote the joint
3. Coronas of Complete Graphs and Null Graphs
It is well known that the corona 0 C2 of Gx and G2 is the graph obtained by taking one copy of Gt (which has fi* vertices) and iij copies of G2, and then joining ith vertex of to every vertex in the ith copy of G2. So the order of G1 0 G2 is n1 + nxn2, where n2 is the order of G2 and its size is m1 4 4 where is the size of Gt
for all i =1,2.
Youssef [15] has shown that Kn 0 is prime if and only if n < 7. In this section, we extend this result to show that Kn 0 K2 is prime if and only if n < 16. Moreover, we show that the graph Km 0 Kn is prime if and only if m = 1 and n = 2, or n = 1 and for all m > 1. Now, we present the proof of above result due to Youssef using our notation.
Lemma 3.1. The graph Kn 0 K1 is prime if and only if n < 7.
Proof. Let n > 8, then n (2rc) < n - 2 and since o.>(ATn 0 = n, we get
o>(Kn 0 Kj.) > n(2n) +1 and hence by theorem 2.1, Kn Q is not prime. Conversely, if n :< 7, then Aj 0 Aj = P2, K2 0 = P4 are trivially prime and
A;: is prime with the following prime labeling (1,2), (3,4), (5,6), i.e., we label the
vertices of as 1,3,5 and we label the corresponding pendent vertices as 2,4,6. For 4 < n < 7, we let V(K,i 0 Ki_) = , ii3, ■■■,u„)U (vi v2 5 v3, ■■■,tn„), where
^(Ar?! 0 Ay) = {u_iuj:l i <j n } { Uj Vj : 1 < i < n ], then 0 Kt, where
4 < n < 7 are prime with the following prime labeling functions: f : V(K4 0 -*
{1,2,3,-.,8] such that f(iij) = 2i-l, f(r,) = 2i, where 1 < i <4 ( i.e., (1,2), (3,4), (5,6), (7,8)), f : V(KS 0 Kx) -» {1,2,3,, ,10] such that f(
= 1,
) = 2,
= 3,
) = 5,
h) = 10,f[!!¿) = 9, f(i;3) = 4, f( r4) = 6, f(rs) = 8 (i.e., (1,10), (2,9), (3,4), (5,6), K6 0 Aj) -> {1,2,3__________12} such that f(iit) = 1, f(»¿) = 2, f(u3) = 3, f(n4) = 5,
(1,10), (2,9), (3,4), (5,6), (7,8),(11,12)) and f : \
7 0 K{1,2,3, ..,14} such that &)=ll,f(«7)=13,f(Vl)=10,f(V2) =
9,
(7,8), (11,12),(13,14)), the lemma follows.
s) = 4, f(„4) = 6, f(i75) = 8, f(i%) = 12, f(r7) = 14 ( i.e., (1,10), (2,9), (3,4), (5,6),
Lemma 3.2. The graph Kn
K2 is prime if and only if n < 16.
bJ[
Proof. Let n > 17, then ^(3n)<n-2 and since ¿u(J J) K?) > rr(3n) + 1 and hence by theorem 2.1, Kn
K?
Conversely, if n < 16, the first n-tuples suffice as a prime labeling for Kn
= ■ ■■, we get is not prime.
D Kz : (1,2,3),
(5,4,6), (7,8,9), (11,10,12), (13,14,15), (17,16,18),(19,20,21), (23,22,24), (26,25,27), (29,28,30), (31,32,33), (35,34,36), (37,38,39), (41,40,42), (43,44,45), (47,46,48), the lemma follows.
Lemma 3.3. The graph Km О Кг is prime for all m > 1.
Proof. Let V{Km 0 K{) = (иг,и2 , u3, U (vi v2 ■> рз, then the
graph 0 Kx is prime with the following prime labeling function
f : V{Km 0 K{) -» (1,2,3,.,,2m] such that f(u() = 2i-l, f(i?j) = 2i, where 1 < i < m (i.e.,(l,2),(3,4),(5,6),---,(2m-l, 2m)), the lemma follows.
Example 3.1. The graph Кг 0 Kz is prime.
Solution. This follows directly since ^0 K2 = K3 and Къ is clearly prime.
Lemma 3.4. If either m > 1 and n > 2, or m > 1 and n = 2, then the graph K-.,- C_', is not prime.
Proof. It is easy to verify that P(Km 0 Kn) = m and m < m1 for all m > 1
and n > 2, or for all m > 1 and n = 2, and hence by theorem 2.1, Km 0 Kn is not prime, the lemma follows.
Theorem 3.1. The graph Km 0 Kn is prime if and only if m = 1 and n = 2, or n = 1 andfor all m > 1.
Proof. The proof follows directly from lemma 3.3, example 3.1 and lemma 3.4, the theorem follows.
4. Joins and Unions of Cycles and Paths
As stated in [9], every path Pn is prime and every cycle Cn is prime. In this section we extend those results to pairs of paths, pairs of cycles, and graphs consisting of one cycle and one path. Deretsky, Lee and Mitchem [2] have shown that the disjoint union ^2k u Cn of two cycles Clk and Cn is prime. The following theorem generalizes this result as follows:
Theorem 4.1. The anion Cn U Cm of two cycles Cn and Cm is not prime if and only if both n and m are odd.
Proof. If both n and m are odd, then it is clear that i>(C_n C m) = /?(C„) + p(Cm) = j + j с and hence by theorem 2.1, the graph
Г;, „1 is not prime. Conversely, we have two cases:
Case 1. n and m are even.
We label the vertices of Cn as 1,2,3, ■■ n, and we label the vertices of C,„ as n+1,
n+2,..., n + m. So, the reader can easily verify that the graph Cn U Cm is prime.
Case 2. n even and m odd ( or vice versa).
We label the vertices of C)t as 2,3, ■ ■ ,n+l, and we label the vertices of Cm as 1, n+2, ■■■, n + m, which clearly give a graph Cn U Cm is prime, the theorem follows.
Youssef [15], has shown that the joint Cn + Cm of two cycles Cn and Cm is not prime for all n > 3 and m > 3 as follows.
Lemma 4.1. The joint Cn + Cm of two cycles Cn and Cm is not prime for all n > 3 and Til > 3.
Proof. Since /?(C_n + C_mJ = max {/?(C_nJ, /?(C_m) }= max { ", y} < —7“ f°r
all n > 3 and m > 3, hence by theorem 2.1, the graph Cn -f Cm is not prime, the lemma follows.
Lemma 4.2. The anion Pt, IJ Pm of two paths P^ and Pin is prime for all n > 1 and 771 > 1.
Proof. Without loss of generality, we assume that n < m, then let K(P_n) = (it 1,1^2 > and =(vi, r2, i?3, ■■■, vm). Therefore the graph U Pm is
prime with the following prime labeling function f: V(Pn U Pm)— + m} such
that f((«j)=i, where 1 < i < n, and f(r;) = j, where n+l<; </1+777, the lemma follows.
Lemma 4.3. The joint Px + Pnof two paths P1 and P„ is prime for all n > 1.
Proof. Let V(Fi)= {it} and V(Pn)= (iiL, u2, u3, ■■■, iiR). Therfore the graph + P„ is prime with the following labeling function f :V(Pi + P„) -• {1,2,3,—,« + 1} such that f(ii) = 1 and f(/Uj)= j, where 2 < j < n + 1, the lemma follows.
Lemma 4.4. The joint P2 + P„ of two paths P2 and P„ is prime for all n > I if and only if n odd.
Proof. If n is even, then it is clear that /?(P_2+ P_n) = max {7, 7-} < —7“ for all
even n > 2, hence the necessity follows directly from theorem 2.1. Conversely, suppose that n is odd, and let V(P2 + Pn)= U [u1,u2,u3,- -,un}, then we have two cases.
Case 1. n+2 is prime.
The graph P2 + Pn is prime with the following prime labeling function f : V(P2 +P„)-> {1,2,3,+ 2} such that f(u) = 1, f(v) = n+2, and f(u,) = i+1, where 1 < i < n.
Case 2. n+2 is not prime.
The graph P2 + Pn is prime with the following prime labeling function f : V(P2 + Pn)-’ {1,2,3, -,n -1- 2} such that f(u) = 1, f(v) = p, where p is the greatest prime less than n+2, f(uO = p+1, i'(u2) = p+2, f(u3) = p+3, ■■■, f(«p_3) = n+2, f(up_2) = 2, ■■■, f(tjf.) = p-1, i.e., we label the vertices of P2 as 1, p and we label the vertices of Pn as p+1, p+2, ■■■, n+2, 2, 3, ■ ■ ■, p-1. Therefore the sufficiency follows, the lemma follows.
Lemma 4.5. The joint Рп + Рт of two paths PK and Pm is not prime for all n > 1 and m > 1 except for n = 1 andfor all m > 1(or vice versa), or n = 2 and m odd ( or vice versa).
Proof. The proof follows directly from the fact that /?(P_n+ Pm) = max {у, у} < for all n > 3 and m > 1 ( or vice versa), or n = 2 and m even ( or vice versa), and theorem 2.1, the lemma follows.
Theorem 4.2. The joint Pn + Pm of two paths Pri and Pm is prime if and only if n =1 andm > l(or vice versa), or n =2 andm odd (or vice versa)
Proof. The proof follows directly from lemma 4.3, lemma 4.4 and lemma 4.5, the theorem follows.
Lemma 4.6. The anion Cn U Pm of cycls Cn and paths Pm is prime for all n > 3 and m > 1.
Proof. Without loss of generality, we assume that n < m, then let V(C_n) = (ul5u2 , i<j, -,«,,) and F(P_m) =(pj, v2, v3, ■■■, гщ). Therefore the graph Cn U Рщ is prime with the following prime labeling function f : V (Cn U Рщ) -* (1,2,3,—,n + 771} such that f(tJt)=i, where 1 < i < n, and f(r;) = j, where n + 1 < j < n + m, the lemma follows.
Lemma 4.7. The joint Cn + Pm of cycles C„ and paths Pm is not prime for all 7i > 3 and m > 1.
Proof. If n = 3 and m < 2, then from the fact that Kn is not prime for all n > 4, + PL = Ki and + P; = К5, we get Cn + Pm is not prime. For n > 3 and m > 3, it is clear that /?(C_n+ P m) = max {(С,;)(Р,?;)]= max < hence by theorem 2.1,
the graph Cn + Pm is not prime, the lemma follows.
5. Complete Tripartite Graphs with Other Graphs
Seoud, Diab, and Elsakhawi [12], have shown the following complete bipartite
graphs are prime: K2 m and К3чГП unless m = 3 or m = 7. In this section, we extend those results to show that is prime for all 7i > 1 and Kx 2itlis prime for all rt > 1 except n = 3 or n = 7. Moreover, we show that the following graphs are prime: Pn U Kxm f°r all n > 1 and in > 1, Cn U Kl m, for all n > 3 and j?i > 1, Kn U KXm for all n > 1 and m > 1, I<! + Kl m for all m >1, Р„ О Kl for all n > 1, Kl О P„ for all n > 1, and Kn U Km for all n > 1 and m > 1.
Lemma 5.1. The complete tripartite graph is prime for all n > 1.
Proof. Let the set of vertices of К1Л n be L = {u}, M ={г?} and N = (иъи2 , ti3, ■■■,7(п), then we label the vertices of К1Л п as u =1, v = p, where p is the greatest prime less than or equal to n+2, and we label the vertices of N by the remaining labels, which give a prime labeling and hence the considered graph is prime, the lemma follows.
Lemma 5.2. The complete tripartite graph if12n is prime for all n > 1 except for n = 3, or n = 7.
Proof. Let the set of vertices of Klm2 n be L = {u}, M = {u, w] and N = (u!,u2 , n3, -,11b), then we label the vertices of Kh2,n as as u = 1, v = p*, w = p^, where p1 < p2 are two greatest prime less than or equal to n+2, and we label the vertices of N by the remaining labels, which give a prime labeling and hence the considered graph is prime, the lemma follows.
Lemma 5.3. The anion P„ U Ki,m °f pciths Pn and stars Kim is prime for all n > 1 and m > 1.
Proof. Without loss of generality, we assume that 1 < m < ti. If m = 1, then it is easy to verify that Pn U Kirl = P^ U P^, and hence by lemma 4.2, we get Pn U K±,i is prime. Now, let m > 1 and the vertex sets of Kl m be L = {uj and M = {itj: 1 < i < m}, and let the vertex set of Pn be N ={vj : 1 < i < n}, then we label the vertices of the sets L and M as u = 1 and u,- = i + 1 for all 1 < i < m, and we label the vertices of the set N by remaining labels, i.e, Vj = j + m for all 2 < j < n. Hence we can easily verify that the considered graph is prime, the lemma follows.
Lemma 5.4. The anion Cn U A'l m of cycles Cn and stars Kl m is prime for all 7i > 3 and Til > 1.
Proof. Without loss of generality, we assume that nt < n. If m = 1, then it is easy to verify that Cn (J Kt.i = U P^, and hence by lemma 4.6, we get C„ U Ki,m is prime. Now, let > 1 and the vertex sets of Klm be L = {u} and M ={Uj: 1 < i < rn}, and let the vertex set of be N ={vj: I < i < n], then we label the vertices of the sets L and N as u = p, where p is the greatest prime number in the set {1,2,3/--,n + m + 1} and Vj = j for all 1 < j < m, and we label the vertex set M by remaining labels. Hence we can easily verify that the considered graph is prime, the lemma follows.
Lemma 5.5. The following graphs are prime:
(a) Kn U Ki.m for all n > 1 and m > 1,
(b) JTj + Kl m for all i ti > 1,
(c) Pn O Kxfor all n > 1,
(d) K± O Pffor all n > 1,
(e) Kn (J Kinf()r all n > 1 and m > 1.
Proof, (a) Without loss of generality, we assume that m < n. Now, let the vertex
sets of Kim be L = {u.} and M ={1/^: 1 < ¡. < m}, and let the vertex set of Kn be N ={Vj : 1 < r. < n} , then we label the vertices of the sets L and M as u =1 and i/i = i for _ , and we label the vertex set N by remaining labels. Hence we can easily verify
that the considered graph is prime.
(b) Let the vertex sets of Kl m be L = {u} and M ={ut: 1 < j < m}, then we label
the vertex of as 1, we label the vertex set L as u = p, where p is the greatest prime
number in the set {1,2,3,—,m + 2}, and we label the vertex set M by remaining labels. Hence we can easily verify that the considered graph is prime.
(c) We label the vertices of Pn as 1, 3, 5, ■■■, 2n-l, and we label the pendent vertices as 2, 4, 6, ■ ■ ■, 2n. Hence we can easily verify that the considered graph is prime.
(d) The proof follows directly from the fact that 0 Pn = Pi + and theorem 4.2.
(e) Without loss of generality, we assume that m < n. Now, we label the vertices of Km as 1, 2, 3,m, and we label the vertices of Kn as m+1, m+2, m+3, ■■■, n + m. Hence we can easily verify that the considered graph is prime, the lemma follows.
6. Acknowledgment.
Part of the work for this paper was done while the author was visiting the Faculty of Mathematics and information technology, Belgorod States University, Belgorod, Russia. The author would like to thank the Faculty of Mathematics and information technology for its hospitality.
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НЕКОТОРЫЕ РЕЗУЛЬТАТЫ О ПЕРВИЧНОЙ МАРКИРОВКЕ ГРАФОВ
АДЕЛЬ Т. ДИАБ
Университет Аин Шамс Факультет естественных наук e-mail:: [email protected]
Граф называется первичным, если он имеет относительтельную первичную маркировку своих граней, удовлетворяющих определенным свойствам. Целью настоящей статьи является описание некоторых новых семейств графов, обладающих первичной маркировкой, в терминах необходимых и достаточных условий.
Ключевые слова: Граф, маркировка граней.
