Abstract
An L(2, 1)-labeling of a graph is an assignment of non-negative integers, called colours to the vertex set of G such that the difference between the colours assigned to adjacent vertices is at least two and the colours assigned to any two vertices at distance two are distinct. The L(2, 1)-labeling number λ2,1(G) of G is the minimum range of label over all such possible labelings. It was shown by Bodlaender et al. (Comput J 47(2):193–204, 2004) that \({\lambda_{2,1}(G)\leq 5\Delta-2}\) , when G is a permutation graph. In this paper, the authors improve the upper bound for permutation graphs to max\({\{4\Delta-2, 5\Delta-8\}}\) , by doing a detailed analysis of Chang and Kuo’s heuristic for L(2, 1)-labeling of general graphs applied to the particular case of permutation graphs. On the other hand, Araki (Discrete Appl Math 157:1677–1686, 2009) showed that, for a bipartite permutation graph G, \({bc(G)\leq \lambda_{2,1}(G)\leq bc(G)+1}\) , where bc(G) is the biclique number of G. This paper also provides sufficient conditions of bipartite permutation graphs G such that \({\lambda_{2,1}(G)=bc(G)+1}\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Araki T.: Labeling bipartite permutation graphs with a condition at distance two. Discrete Appl. Math. 157, 1677–1686 (2009)
Bertossi, A.A., Pinotti, C.M., Tan, R.B.: Efficient use of radio spectrum in wireless networks with channel separation between close stations. In: DIALM 2000 Proceedings of the 4th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, pp. 18–27 (2000)
Bodlaender H.L., Kloks T., Tan R.B., Leeuwen J.V.: Approximations for λ-colorings of graphs. Comput. J. 47(2), 193–204 (2004)
Bonomo F., Cerioli M.R.: On the L(2, 1)-labelling of block graphs. Int. J. Comput. Math. 88(3), 468–475 (2011)
Calamoneri T., Caminiti S., Olariu S., Petreschi R.: On the L(h, k)-labeling of co-comparability graphs and circular-arc graph. Networks 53(1), 27–34 (2009)
Calamoneri T.: The L(h, k)-labelling problem: an updated survey and annotated bibliography. Comput. J. 54(8), 1344–1371 (2011)
Calamoneri T., Petreschi R.: L(2, 1)-Labeling of unigraphs. Discrete Appl. Math. 159(12), 1196–1206 (2011)
Calamoneri T., Petreschi R.: L(h, 1)-Labeling subclasses of planar graphs. J. Parallel Distrib. Comput. 64(3), 414–426 (2004)
Cerioli M.R., Posner D.F.D.: On L(2, 1)-coloring split, chordal bipartite, and weakly chordal graphs. Discrete Appl. Math. 160, 2655–2661 (2012)
Chang G.J., Kuo D.: The L(2, 1)-labeling on graphs. SIAM J. Discrete Math. 9, 309–316 (1996)
Corneil D.G., Perl Y., Stewart L.K.: A linear recognition algorithm for cographs. SIAM J. Comput. 14, 926–934 (1985)
Fiala J., Kloks T., Kratochvil J.: Fixed-parameter complexity of λ-labelings. Discrete Appl. Math. 113, 59–72 (2001)
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. 2nd edn. Elsevier, Amsterdam (2004)
Goncalves D.: On the L(d, 1)-labellinng of graphs. Discrete Math. 308, 1405–1414 (2008)
Griggs J., Yeh R.K.: Labeling graphs with a condition at distance two. SIAM J. Discrete Math. 5, 586–595 (1992)
Hasunuma, T., Ishii, T., Ono, H., Uno, Y.: A linear time algorithm for L(2, 1)-labeling of trees. In: Lecture Notes in Computer Science, vol. 5757, pp. 35–46 (2009)
Havet, F., Reed, B., Sereni, J. S.: L(2, 1)-labeling of graphs. In: Proceedings of 19th annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, SIAM, pp. 621–630 (2008)
Khan N., Pal A., Pal M.: L(2, 1)-Labelling of cactus graphs. Mapana J. Sci. 11(4), 15–42 (2012)
Khan N., Pal M., Pal A.: (2, 1)-Total labelling of cactus graphs. J. Inf. Comput. Sci. 5(4), 243–260 (2010)
Khan N., Pal M., Pal A.: L(0,1)-Labelling of cactus graphs. Commun. Netw. 4, 18–29 (2012)
Kohl A.: Bounds for the L(d, 1): number of diameter 2 graphs, trees and cacti. Int. J. Mob. Netw. Des. Innov. 1(2), 124–135 (2006)
Král’ D., Škrekovski R.: A theory about channel assignment problem. SIAM J. Discrete Math. 16, 426–437 (2003)
Král’ D.: Coloring powers of chordal graphs. SIAM J. Discrete Math. 18(3), 426–437 (2005)
Lai T.H., Wei S.S.: Bipartite permutation graphs with application to the minimum buffer size problem. Discrete Appl. Math. 74, 33–55 (1997)
Pal M., Bhattacharjee G.P.: A data structure on interval graphs and its applications. J. Circuits Syst. Comput. 7(3), 165–175 (1997)
Panda B.S., Goel P.: L(2, 1)-Labeling of perfect elimination bipartite graphs. Discrete Appl. Math. 159, 1878–1888 (2011)
Paul, S., Pal, M., Pal, A.: L(2, 1)-Labelling of interval and circular-arc graphs (communicated)
Paul S., Pal M., Pal A.: An efficient algorithm to solve L(0,1)-labelling problem on interval graphs. Adv. Model. Optim. 15(1), 31–43 (2013)
Roberts F.S.: T-Colorings of graphs: recent results and open problems. Discrete Math. 93, 229–245 (1991)
Saha A., Pal M., Pal T.K.: Selection of programme slots of television channels for giving advertisement: a graph theoretic approach. Inf. Sci. 177(12), 2480–2492 (2007)
Sakai D.: Labeling chordal graph: distance two condition. SIAM J. Discrete Math. 7, 133–140 (1994)
Spinrad J., Brandstdt A., Stewart L.: Bipartite permutation graphs. Discrete Appl. Math. 18(3), 279–292 (1987)
Uehara R., Valiente G.: Linear structure of bipartite permutation graphs and the longest path problem. Inf. Process. Lett. 103, 71–77 (2007)
Yeh R.K.: A survey on labeling graphs with a condition at distance two. Discrete Math. 306, 1217–1231 (2006)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Paul, S., Pal, M. & Pal, A. L(2, 1)-Labeling of Permutation and Bipartite Permutation Graphs. Math.Comput.Sci. 9, 113–123 (2015). https://doi.org/10.1007/s11786-014-0180-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-014-0180-2