On the Prime Labeling of Generalized Petersen Graphs P (n, 3) 1

Int. J. Contemp. Math. Sciences, Vol., 0, no., - 00 On the Prime Labeling of Generalized Petersen Graphs P (n, ) Kh. Md. Mominul Haque Department of Computer Science and Engineering Shahjalal University of Science and Technology Sylhet-, Bangladesh momin@gmail.com Lin Xiaohui, Yang Yuansheng and Zhao Pingzhong Department of Computer Science and Engineering Dalian University of Technology Dalian, 0, P. R. China Abstract A graph G with vertex set V is said to have a prime labeling if its vertices can be labeled with distinct integers,,..., V such that for every edge xy in E, the labels assigned to x and y are relatively prime or coprime. A graph is called prime if it has a prime labeling. In this paper, we show that generalized Petersen graphs P (n, ) are not prime for odd n, prime for even n 00 and conjectured that P (n, ) are prime for all even n. Mathematics Subject Classification: Primary 0C; Secondary C0 Keywords: Prime labeling, Prime graph, generalized Petersen graph Introduction We consider only finite undirected graphs without loops or multiple edges. G =(V,E) be a graph with vertex set V and edge set E. A graph G is said to have a prime labeling if its vertices can be labeled with distinct integers,,..., V such that for every edge xy in E, the labels assigned to x and y are relatively prime or coprime. A graph is called prime The research is supported by Chinese Natural Science Foundations (00).

Kh. Md. Mominul Haque et al if it has a prime labeling. This concept was originated with Entringer and introduced by Tout, Dabboucy, and Howalla[0]. Roger Entringer conjectured that all trees are prime. Fu and Huang [] proved that every tree with n vertices is prime. O Pikhurko [, ] extended this result to all n 0. Other prime graphs include all cycles and the disjoint union of C k and C n. Seoud, Diab, and Elsakhawi [] showed that following graphs are prime: Fans, Helms, Flowers, Stars, K,n and K,n unless n = or. They also showed that P n + K m (m ) is not prime. Kelli Carlson [] proved that generalized Books and C m -Snakes are prime graphs. Vilfred, Somasundaram, and Nicholas [] have conjectured that the grid P m P n is prime when n is prime and n>m. This conjecture was proved by Sundaram, Ponraj, and Somasundaram []. In the same article they also showed that P n P n is prime when n is prime. The authors [, ] proved that the following graphs are prime: generalized Petersen graph P (n, ) for even n 00 and not prime for odd n and Knödel graphs W,n for n 0. We refer the readers to the dynamic survey by Gallian []. T he generalized P etersen graphs P (n, k) are defined to be a graph on n(n ) vertices with V (P (n, k)) = {v i,u i :0 i n } and E(P (n, k)) = {v i v i+,v i u i,u i u i+k :0 i n, subscripts modulo n}. In this paper, we show that generalized Petersen graphs P (n, ) are not prime for odd n, prime for even n 00, and conjecture that P (n, ) are prime for all even n. Main Results Theorem.. P (n, ) is not prime for odd n. By contradiction. Suppose that P (n, ) is prime for some odd n, sayn. f be a prime labeling of P (n, ). Then one of {f(v 0 ),f(v ),...,f(v n )} and {f(u 0 ),f(u ),...,f(u n )} must contains at least n + evens, i.e. there are at least two evens adjacent, a contradiction. For even n, let N i = {n : n + i is prime}, Ni = {n :n + i is prime}, N = i (N i+ Ni+ Ni+ ), N = i (N i+ N i+ N i+ ). We will prove the following Theorem by Lemmas. -.. Theorem.. P (n, ) is prime for even n N N. Observation.. f(u) and f(v) are coprime if they satisfy any one of the following conditions:

On the prime labeling of generalized Petersen graphs P (n, ) () f(u) =orf(v) =, () f(u) = and f(v) is odd or f(v) = and f(u) is odd, () f(u)+f(v) is prime, () f(u) f(v) =, () f(u) f(v) =, () f(u) f(v) = p t p t...p t k k and f(v) 0modp i ( i k), () f(u) f(v) = p t is a prime power and f(u) 0modp. Lemma.. P (n, ) is prime for even n N. i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n +, i = n, i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n +, i = n, n +, i = n, n, i = n. 0 Figure : P (n, ) for n = N. In Figure., we show the prime labeling of P (n, ), where n = N. For 0 i n, by Observation.(), f(v i ) and f(v i+ ) are coprime. For n i n, f(v i+ ) f(v i ) =, by Observation.(), f(v i ) and f(v i+ ) are coprime. For i = n, f(v 0 ) = and f(v n )=n + is odd, by Observation.(), f(v n ) and f(v 0 ) are coprime. For 0 i n, by Observation.(), f(u i ) and f(u i+ ) are coprime. For i = n, f(u n ) f(u n ) = n (n +) =, by Observation.(), f(u n ) and f(u n ) are coprime. For i = n, f(u n ) f(u n ) = n + (n +) =, by Observation.(), f(u n ) and f(u n ) are coprime. For i = n, f(u n ) f(u n ) = n (n ) =. Since n N, n mod 0. Since n is odd, we have f(u n ) and f(u n ) are coprime. For i = n, by Observation.(), f(u n )=n+ and

Kh. Md. Mominul Haque et al f(u 0 ) = are coprime. For i = n, f(u n ) f(u n ) = n+ (n+) =, by Observation.(), f(u n ) and f(u n ) are coprime. For i = n, f(u n )=,f(u n )=n. Since n N N N,nmod 0, hence f(u n ) and f(u n ) are coprime. For 0 i n, { i + (i +) =, i mod = 0, f(u i ) f(v i ) = n i (n i) =, i mod =, by Observation.(), f(v i ) and f(u i ) are coprime. For i = n, f(u n ) f(v n ) = n+ (n ) =, by Observation.(), f(v n ) and f(u n ) are coprime. For i = n, f(u n ) f(v n ) = n + n =, by Observation.(), f(v n ) and f(u n ) are coprime. For i = n, f(u n ) f(v n ) = n (n +) = and f(v n ) is odd, by Observation.() f(v n ), f(u n ) are comprime. Hence f is a prime labeling of P (n, ) for even n N. For the Lemmas. -., we only define f, and leave for the readers to verify that the f is a prime labeling of P (n, ). Lemma.. P (n, ) is prime for even n N. i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n =0 N. Lemma.. P (n, ) is prime for even n N. i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, i +, 0 i n,i mod = 0, n i, i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n. In Figure., we show the prime labeling of P (n, ), where n = N. Lemma.. P (n, ) is prime for even n N.

On the prime labeling of generalized Petersen graphs P (n, ) 0 0 0 0 P(0, ), 0 N N P(, ), N N N (a) Figure : i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n, n, i = n, n, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N. 0 0 0 0 0 P(, ), N N P(, ), N N N (a) Figure : Lemma.. P (n, ) is prime for even n N.

Kh. Md. Mominul Haque et al i +, 0 i n,i mod = 0, n i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, i +, 0 i n 0,i mod = 0, n i, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n, n, i = n, n, i = n. In Figure., we show the prime labeling of P (n, ), where n = N. Lemma.. P (n, ) is prime for even n N. +i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n, +i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n, n, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N. 0 0 0 0 0 0 P(, ), N N P(, ), N N N (a) Figure : Lemma.0. P (n, ) is prime for even n N. In N, there is only one integer smaller than, namely. Since N, by Lemma., P (, ) is prime. Hence, we only consider even n. And we define the function f as follows:

On the prime labeling of generalized Petersen graphs P (n, ) 0 + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n, +i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n, n, i = n. In Figure., we show the prime labeling of P (n, ), where n = N. Lemma.. P (n, ) is prime for even n N. + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n, 0, i = n,, i = n,, i = n,, i = n, + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, n, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N. 0 0 0 0 P(, ), N N P(, ), N N N (a) Figure : Lemma.. P (n, ) is prime for even n N.

0 Kh. Md. Mominul Haque et al + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n, 0, i = n,, i = n,, i = n,, i = n, + i, 0 i n 0,i mod = 0, n i, 0 i n 0,i mod =,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, n, i = n,, i = n,, i = n. In Figure., we show the prime labeling of P (n, ), where n = N. Lemma.. P (n, ) is prime for even n N. In N, there is only one integer smaller than 0, namely 0. Since 0 N, by Lemma., P (0, ) is prime. Hence, we only consider even n 0. And we define the function f as follows: + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n 0,, i = n, 0, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n 0,, i = n,, i = n, n, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, 0, i = n. Lemma.. P (n, ) is prime for even n N. In N, there is only one integer smaller than, namely. Since N, by Lemma., P (, ) is prime. Hence, we only consider even n. And we define the function f as follows: + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n, 0, i = n 0,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, 0, i = n,, i = n, + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n 0,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, n, i = n.

On the prime labeling of generalized Petersen graphs P (n, ) Lemma.. P (n, ) is prime for even n N. In N, there is only one integer smaller than, namely. Since N, by Lemma., P (, ) is prime. Hence, we only consider even n. And we define the function f as follows: 0 + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n,, i = n 0,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, + i, 0 i n,i mod = 0, n i, 0 i n,i mod =,, i = n,, i = n,, i = n,, i = n, n, i = n,, i = n 0, 0, i = n,, i = n,, i = n,, i = n,, i = n,, i = n, 0, i = n,, i = n,, i = n. In Figure.(a) we show the prime labeling of P (n, ) for even n = N. 0 0 0 0 0 0 0 0 0 P(, ), N N P(0, ), 0 N N N (a) Figure : Lemma.. P (n, ) is prime for even n N N N. i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n +, i = n, n, i = n, n, i = n, i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n +, i = n, n, n, n, i = n.

Kh. Md. Mominul Haque et al In Figure. we show the prime labeling of P (n, ) for even n =0 N N N. Lemma.. P (n, ) is prime for even n N N N. i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n, i = n, n, i = n, i +, 0 i n,i mod = 0, n + i, 0 i n,i mod =, n, i = n, n, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N N N. 0 0 0 P(, ), N N N P(, ), N N N (a) Figure : Lemma.. P (n, ) is prime for even n N N N. n + i, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n + i, 0 i n,i mod = 0, i +, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n. In Figure., we show the prime labeling of P (n, ), where n = N N N. Lemma.. P (n, ) is prime for even n N N N. In N N N, there is only one integer smaller than, namely. Since N N N, by Lemma., P (, ) is prime. Hence, we only consider even n. And we define the function f as follows:

On the prime labeling of generalized Petersen graphs P (n, ) n + i, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n + i, 0 i n,i mod = 0, i +, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N N N. Lemma.0. P (n, ) is prime for even n N N N. n + i, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n + i, 0 i n,i mod = 0, i +, 0 i n,i mod =, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n, n, i = n, n, i = n. In Figure., we show the prime labeling of P (n, ), where n = N N N. 0 0 0 0 0 0 0 P(, ), N N N P(, ), N N N (a) Figure : Lemma.. P (n, ) is prime for even n N N N. In N N N, there is only one integer smaller than, namely 0. Since 0 N N N, by Lemma., P (0, ) is prime. Hence, we only consider even n. And we define the function f as follows: Case. n mod.

Kh. Md. Mominul Haque et al n + i, 0 i n 0,i mod = 0, i, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n + i, 0 i n 0,i mod = 0, i +, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n. Case. n mod. n + i, 0 i n 0,i mod = 0, i, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n + i, 0 i n 0,i mod = 0, i +, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n. Case. n mod. n + i, 0 i n 0,i mod = 0, i, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n + i, 0 i n 0,i mod = 0, i +, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n. Case. n,, mod. n + i, 0 i n 0,i mod = 0, i, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n + i, 0 i n 0,i mod = 0, i +, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N N N.

On the prime labeling of generalized Petersen graphs P (n, ) Lemma.. P (n, ) is prime for even n N N N. In N N N, there are only two integers smaller than, namely,. Since N N N, by Lemma., P (, ) is prime. Since N, by Lemma., P (, ) is prime. Hence, we only consider even n. And we define the function f as follows: Case. n 0 mod. n + i 0, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n 0, i = n, n, i = n, n + i, 0 i n,i mod = 0, i +, i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n. Case. n mod. n i 0, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n 0, i = n, n, i = n, n i, 0 i n,i mod = 0, i +, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n. Case. n mod. n i 0, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n 0, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n i, 0 i n,i mod = 0, i +, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n. Case. n 0,, mod, and n.

Kh. Md. Mominul Haque et al n + i 0, 0 i n,i mod = 0, i, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n 0, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n i, 0 i n,i mod = 0, i +, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n 0, i = n, n, i = n, n, i = n. 0 0 0 0 0 0 P(, ), N N N P(, ), N N N (a) Figure : Lemma.. P (n, ) is prime for even n N N N. i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n, i +, 0 i n,i mod = 0, i + n +, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n. In Figure., we show the prime labeling of P (n, ), where n = N N N. Lemma.. P (n, ) is prime for even n N N N.

On the prime labeling of generalized Petersen graphs P (n, ) i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n, i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n +, i = n, n +, i = n, n, i = n, n +, i = n, n +, i = n. In Figure.0(a), we show the prime labeling of P (n, ), where n = N N N. 0 0 0 0 P(, ), N N N P(0, ), 0 N N N (a) Figure 0: Lemma.. P (n, ) is prime for even n N N N. In N N N, there is only one integer smaller than 0, namely. Since N N N, by Lemma., P (, ) is prime. Hence, we only consider even n 0. And we define the function f as follows: i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n, i = n, n, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n, n, i = n, i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n, i = n. In Figure.0, we show the prime labeling of P (n, ), where n =0 N N N. Lemma.. P (n, ) is prime for even n N N N. In N N N, there is only one integer smaller than, namely. Since N N N, by Lemma., P (, ) is prime. Hence, we only consider even n. And we define the function f as follows:

Kh. Md. Mominul Haque et al Case. n mod. i +, 0 i n 0,i mod = 0, n + i +, 0 i n 0,i mod =, n, i = n, n, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n, i = n, n, i = n, i +, 0 i n 0,i mod = 0, n + i +0, 0 i n 0,i mod =, n, i = n, n, i = n, n +, i = n, n, i = n, n +, i = n, n +0, i = n, n +, i = n, n +, i = n, n, i = n. Case. n mod. i +, 0 i n 0,i mod = 0, n + i +, 0 i n 0,i mod =, n, i = n, n +, i = n, n +, i = n, n, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n, n +, i = n, i +, 0 i n 0,i mod = 0, n + i +0, 0 i n 0,i mod =, n, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n, n, i = n, n +, i = n, n +0, i = n, n +, i = n. In Figure.(a), we show the prime labeling of P (n, ), where n = N N N. 0 0 0 0 0 0 0 0 0 0 0 0 P(, ), N N N P(0, ), 0 N N N (a) Figure : Lemma.. P (n, ) is prime for even n N N N. In N N N, there is only one integer smaller than 0, namely. Since N N N, by Lemma., P (, ) is prime. Hence, we only consider even n 0. And we define the function f as follows:

On the prime labeling of generalized Petersen graphs P (n, ) Table.. T T T T T T T T n N i Ni+ Ni+ n N i Ni+ Ni+ n N i Ni+ Ni+ n N i Ni+ Ni+ T T T T N N N 0 N N N N T T T T N N N N N T T T T N N N N N N N T T N N N T T N N T T T T T T N N N N N N 0 N N N N T T N N T T T T T T 0 N N N N N N N N N N T T N N T T T T T T T T N T N T N N T N T N N T N T N 0 N T N T N N T N T N 0 N T N T N N T N T N N T N T N N T N T N N T N T N N T N T N N T N T N N T N T N N T N T N 0 N T N T N N T N T N 0 N T N T N N T N T N N T N T N N T N T N N T N T N N T N T N N T N T N 00 N N N N N N 0 N N N N N N N N N N N N N N N N N N N N N 0 N N N i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n, i = n, n, i = n 0, n, i = n, n, i = n, n, i = n, n +, i = n, n, i = n, n, i = n, n +, i = n, n +, i = n, n +0, i = n, i +, 0 i n,i mod = 0, n + i +, 0 i n,i mod =, n, i = n, n +, i = n 0, n, i = n, n, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n, n +, i = n. In Figure., we show the prime labeling of P (n, ), where n =0 N N N. From the Lemmas. -., Theorem. holds. Furthermore, we have the following conjecture Conjecture.. P (n, ) is prime for all even n. Since n N N for any even n 00, by Theorem. and Table., we have Conjecture. holds for even n 00. References [] K. Carlson, Generalized books and C m -snaks are prime graphs, Ars Combinatora 0(00),. [] H. L. Fu and K. C. Huang, On Prime labellings, Discrete Math. (),. [] J. A. Gallian, A Survey: A dynamic survey of graph labeling, Electronic Journal of Combinatorics. (00), -.

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