Abstract
L(0,1)-labelling of a graph G=(V,E) is a function f from the vertex set V(G) to the set of non-negative integers such that adjacent vertices get number zero apart, and vertices at distance two get distinct numbers. The goal of L(0,1)-labelling problem is to produce a legal labelling that minimize the largest label used. In this article, it is shown that, for a permutation graph G with maximum vertex degree Δ, the upper bound of λ 0,1(G) is Δ−1. Finally, we prove that the result is exact for bipartite permutation graph.
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Barman, S.C., Mondal, S., Pal, M.: An efficient algorithm to find next-to-shortest path on permutation graph. J. Appl. Math. Comput. 31(1-2), 369–384 (2009)
Bertossi, A.A., Bonuccelli, M.A.: Code assignment for hidden terminal interference avoidance in multihope packet radio networks. IEEE/ACM Trans. Networking 3(4), 441–449 (1995)
Bodlaender, H.L., Kloks, T., Tan, R.B., Leeuwen, J.V.: Approximations for λ-colorings of graphs. Comput. J. 47(2), 193–204 (2004)
Calamoneri, T., Petreschi, R.: L(h,1)-labeling subclasses of planar graphs. J. Parallel Distrib. Comput. 64(3), 414–426 (2004)
Calamoneri, T.: The L(h,k)-labelling problem: An updated survey and annotated bibliography. Comput. J. 54(8), 1344–1371 (2011)
Chang, G.J., Lu, C.: Distance two labelling of graphs. Eur. J. Comb. 24, 53–58 (2003)
Golumbic, M.C.: Algorithmic graph theory and perfect graphs, 2nd edn. Elsevier (2004)
Goncalves, D.: On the L(d,1)-labellinng of graphs. Discret. 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)
Hale, W.K.: Frequency assignment: theory and applications. Proc. IEEE 68, 1497–1514 (1980)
Jin, X.T., Yeh, R.K.: Graph distance-dependent labelling related to code assignment in compute networks. Nav. Res. Logist. 51, 159–164 (2004)
Khan, N., Pal, M., Pal, A.: (2,1)-total labelling of cactus graphs. Int. J. Inf. Comput. Sci. 5(4), 243–260 (2010)
Khan, N., Pal, A., Pal, M.: Labelling of cactus graphs. Mapana Journal of Sciences 11(4), 15–42 (2012)
Khan, N., Pal, A., Pal, M.: L(0,1)-labelling of cactus graphs. Commun. Netw. 4, 18–29 (2012)
Lai, T.H., Wei, S.S.: Bipartite permutation graphs with application to the minimum buffer size problem. Discret. Appl. Math. 74, 33–55 (1997)
Makansi, T.: Transmitter-oriented code assignment for multihop packet radio. IEEE Trans. Commun. 35(12), 1379–1382 (1987)
Mondal, S., Pal, M., Pal, T.K.: An optimal algorithm for finding depth-first spanning tree on permutation graphs. Korean J. Computation and Appl. Maths 6(3), 493–500 (1999)
Mondal, S., Pal, M., Pal, T.K.: An optimal algorithm to solve all-pairs shortest paths problem on permutation graphs. J. Mathematical Modelling and Algorithms 2(1), 57–65 (2003)
Mondal, S., Pal, M., Pal, T.K.: Optimal sequential and parallel algorithm to compute a Steiner tree on permutation graphs. Int. J. Comput. Math. 80(8), 937–943 (2003)
Pal, M.: Efficient algorithm to compute all articulation points of a permutation graphs. J. Computational and Applied Mathematics 5(1), 141–152 (1998)
Pal, M.: A parallem algorithm to generate all maximal independent sets on permutation graphs. Int. J. Comput. Math. 67(3-4), 261–274 (1998)
Paul, S., Pal, M., Pal, A.: An efficient algorithm to solve L(0,1)-labelling problem on interval graphs. Advanced Modeling and Optimization 15(1), 31–43 (2013)
Paul, S., Pal, M., Pal, A.: L(2,1)-labeling of permutation and bipartite permutation graphs. Math. Comput. Sci. 9, 113–123 (2015)
Paul, S., Pal, M., Pal, A.: L(2,1)-labeling of interval graphs. J. Appl. Math. Comput. doi:10.1007/s12190-014-0846-6
Saha, A., Pal, M., Pal, T.K.: Maximum weight k-independent set problem on permutation graphs. Int. J. Comput. Math. 80(12), 1477–1487 (2003)
Saha, A., Pal, M., Pal, T.K.: An efficient PRAM algorithm for maximum-weight independent set on permutation graphs. J. Appl. Math. Comput. 19(1-2), 77–92 (2005)
Spinrad, J., Brandstdt, A., Stewart, L.: Bipartite permutation graphs. Discret. Appl. Math. 18(3), 279–292 (1987)
Yeh, R.K.: A survey on labeling graphs with a condition at distance two. Discret. Math. 306, 1217–1231 (2006)
Zhang, S., Ma, Q.: Labelling of some planar graphs with a condition at distance two. J. Appl. Math. Comput. 24, 421–426 (2007)
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Paul, S., Pal, M. & Pal, A. L(0,1)-labelling of Permutation Graphs. J Math Model Algor 14, 469–479 (2015). https://doi.org/10.1007/s10852-015-9280-5
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DOI: https://doi.org/10.1007/s10852-015-9280-5