SciELO - Scientific Electronic Library Online

vol.33 número4Companions of Hermite-Hadamard Inequality for Convex Functions (II)The upper open monophonic number of a graph índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Servicios Personalizados




Links relacionados

  • En proceso de indezaciónCitado por Google
  • No hay articulos similaresSimilares en SciELO
  • En proceso de indezaciónSimilares en Google


Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) vol.33 no.4 Antofagasta dic. 2014 

L(1,1)-Labeling of Direct Product of any Path and cycle

Deborah Olayide Ajayi

University of Ibadan

Charles Adefokun

Crawford University



Suppose that [n] = {0, 1, 2,...,n} is a set of non-negative integers and h,k G [n].The L (h, k)-labeling of graph G is the function l : V(G) — [n] such that |l(u) — l(v)| > h if the distance d(u,v) between u and v is 1 and |l(u) — l(v)| > k if d(u,v) = 2. Let L(V(G)) = {l(v): v G V(G)} and let p be the maximum value of L(V(G)). Then p is called Xi^—number of G if p is the least possible member of [n] such that G maintains an L(h, k) — labeling. In this paper, we establish X} — numbers of Pm X Pn and Pm X Cn graphs for all m,n > 2.

Keywords : L(1,1)-labeling, D-2 Coloring, Direct Product of Graphs, Cross Product ofGraphs, Path and Cycle.

Mathematics Subject Classification : Primary: 05C78.


[1] Bertossi, A.A.andBonuccelli, M.A.CodeAssignmentforHidden Terminal Interference Avoidance in multiloop Pocket Radio Networks.IEEE/ACM Trans. on Networking 3,(4), pp. 441-449, (1995).

[2] Berttiti, R., Bertossi, A. A. and Bonuccelli, M. A. Assigning Codes in Wireless networks: Bounds and Scaling Properties. Wirel. Netw., 5, pp. 441-449, (1999).

[3] Calamonerri, T. The L(h, k)-Labeling Problem: A Updated Survey and Annoted Bibliography. The Computer Journal 54(8),pp.1344 —1371, (2011)

[4] Calamonerri, T., Pelc, A., and Petreschi, R. Labeling Trees with a Condition at Distance Two. Discrete Math. 306, pp. 1534-1539, (2006).

[5] Chang, G. J. and Kuo, D. The L(2,1)—Labeling of Graphs SIAM J.Discrete math. 9, pp. 309-316, (1996). [6] Chang, G. J., Ke, W.-T, Kuo, D. A., Liu, D. D.-F and Yeh, R. K.

(d, 1)— Labeling of Graphs. Discrete Math. 220, pp. 57-66, (2000). [7] Chiang,S.-H and Yan, J.-H. On 1)—labeling of Cartesian product of a cycle and a path. Discrete App. Math 156, pp. 2867-2881, (2008).

[8] Georges, J. P, and Mauro, D.W. Generalized Vertex Labeling with a Condition at Distance Two. Congr. Numer., 109, pp. 141-159, (1995).

[9] Georges, J. P, Mauro, D. W. and Stein, M. I. Labeling Products of Complete Graphs with a Condition at Distance Two. SIAM J. Discrete Math. 14, pp. 28-35, (2000).

[10] Georges, J. P, Mauro, D. W. On Regular Graphs Optimally labeled with a Condition at Distance Two. SIAM J. Discrete Math. 17 (2), pp. 320-331, (2003).

[11] Goncalves, D. L(p — 1)—Labellings of Graphs. Discrete Math. 308, pp. 1405-1414, (2008).

[12] Gravier, S., Klavzer, S. and Mollard M., Codes and L(2,1)—Labeling in Sierpinski Graphs. Taiwan. J. math. 4, pp. 671-681, (2004).

[13] Griggs, J. R., Yeh, R. K. Labeling Graph with a condition at Distance Two. SIAM J. Discrete Math. 5, pp. 586-595, (1992).

[14] Hale, W. K. Frequency Assignment: Theory and Application. Proc. IEEE 68, pp. 1497-1514, (1980).

[15] Jha, P. K, Klavzer, S. and Vessel, A. L(2,1)—Labeling of Direct Product of Paths and Cycles. Discrete Applied Math. 145 (2), pp. 141-159, (2005).

[16] Kral', D. and Skrekovski, R. A. Theorem About the Channel Assignment J. Discrete Math. 16 (3), pp. 426-437, (2003).

[17] Kuo, D. and Yan, J.-H. On L(2,1)—Labeling of Cartesian Product of Two Cycles. Discrete Math. 283, pp. 137-144, (2004).

[18] Sakai, T. D. Chordal Graphs with a Condition at Distance Two. SIAM J. Discrete Math. 7, pp. 133-140, (1994).

[19] Schwartz, C. and Sakai, T. D. L(2,1)— Labeling of Product of Two Cycles. Discrete Applied Math. 154, pp. 1522-1540, (2006).

[20] Sopena, E., Wu, J. Coloring the square of the Cartesian product of two Cycles Graphs with a Condition at Distance Two. Discrete Math. 310, pp. 2327-2333, (2010).

[21] Wensong, D. and Peter C.,L. Distance Two Labeling and Direct Product of Graphs. Discrete Math. 308, pp. 3805-3815, (2008).

[22] Weichsel, P. M. The Kronecker Product of Graphs. Proc. Amer. Math. Soc. 13 1962; 47 — 62

[23] Whittlessey, M. A, Georges, J. P. and Mauro, D. W. On the λ—number of Qn and Related Graphs. SIAM J. Discrete Math. 8, pp. 499-506,(1995).


Deborah Olayide Ajayi

Department of Mathematics University of Ibadan, Ibadan, Nigeria

e-mail :


Deborah Olayide Ajayi and Charles Adefokun


Tayo Charles Adefokun

Department of Computer and Mathematical Sciences,

Crawford University,


e-mail :

Received : November 2013. Accepted : September 2014

Creative Commons License Todo el contenido de esta revista, excepto dónde está identificado, está bajo una Licencia Creative Commons