Browse > Article
http://dx.doi.org/10.4134/BKMS.b210153

MORE RELATIONS BETWEEN λ-LABELING AND HAMILTONIAN PATHS WITH EMPHASIS ON LINE GRAPH OF BIPARTITE MULTIGRAPHS  

Zaker, Manouchehr (Department of Mathematics Institute for Advanced Studies in Basic Sciences and School of Computer Science Institute for Research in Fundamental Sciences (IPM))
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.1, 2022 , pp. 119-139 More about this Journal
Abstract
This paper deals with the λ-labeling and L(2, 1)-coloring of simple graphs. A λ-labeling of a graph G is any labeling of the vertices of G with different labels such that any two adjacent vertices receive labels which differ at least two. Also an L(2, 1)-coloring of G is any labeling of the vertices of G such that any two adjacent vertices receive labels which differ at least two and any two vertices with distance two receive distinct labels. Assume that a partial λ-labeling f is given in a graph G. A general question is whether f can be extended to a λ-labeling of G. We show that the extension is feasible if and only if a Hamiltonian path consistent with some distance constraints exists in the complement of G. Then we consider line graph of bipartite multigraphs and determine the minimum number of labels in L(2, 1)-coloring and λ-labeling of these graphs. In fact we obtain easily computable formulas for the path covering number and the maximum path of the complement of these graphs. We obtain a polynomial time algorithm which generates all Hamiltonian paths in the related graphs. A special case is the Cartesian product graph Kn☐Kn and the generation of λ-squares.
Keywords
${\lambda}$-labeling; L(2, 1)-coloring; Hamiltonian path; toughness; bipartite multigraphs;
Citations & Related Records
연도 인용수 순위
  • Reference
1 H. L. Bodlaender, H. Broersma, F. V. Fomin, A. V. Pyatkin, and G. J. Woeginger, Radio labeling with preassigned frequencies, SIAM J. Optim. 15 (2004), no. 1, 1-16. https://doi.org/10.1137/S1052623402410181   DOI
2 H. L. Bodlaender, T. Kloks, R. B. Tan, and J. van Leeuwen, λ-coloring of graphs, in STACS 2000 (Lille), 395-406, Lecture Notes in Comput. Sci., 1770, Springer, Berlin, 2000. https://doi.org/10.1007/3-540-46541-3_33   DOI
3 J. S. Deogun, D. Kratsch, and G. Steiner, 1-tough cocomparability graphs are Hamiltonian, Discrete Math. 170 (1997), no. 1-3, 99-106. https://doi.org/10.1016/0012-365X(95)00359-5   DOI
4 J. Fiala, T. Kloks, and J. Kratochvil, Fixed-parameter complexity of λ-labelings, Discrete Appl. Math. 113 (2001), no. 1, 59-72. https://doi.org/10.1016/S0166-218X(00)00387-5   DOI
5 D. A. Fotakis, S. E. Nikoletseas, V. G. Papadopoulou, and P. G. Spirakis, Hardness results and efficient approximations for frequency assignment problems: radio labelling and radio coloring, Comput. Inform. 20 (2001), no. 2, 121-180.
6 U. Sarkar and A. Adhikari, On characterizing radio k-coloring problem by path covering problem, Discrete Math. 338 (2015), no. 4, 615-620. https://doi.org/10.1016/j.disc.2014.11.014   DOI
7 C. Schwarz and D. S. Troxell, L(2, 1)-labelings of Cartesian products of two cycles, Discrete Appl. Math. 154 (2006), no. 10, 1522-1540. https://doi.org/10.1016/j.dam.2005.12.006   DOI
8 Z. Shao, R. K. Yeh, and D. Zhang, The L(2, 1)-labeling on graphs and the frequency assignment problem, Appl. Math. Lett. 21 (2008), no. 1, 37-41. https://doi.org/10.1016/j.aml.2006.08.029   DOI
9 J. P. Georges, D. W. Mauro, and M. A. Whittlesey, Relating path coverings to vertex labellings with a condition at distance two, Discrete Math. 135 (1994), no. 1-3, 103-111. https://doi.org/10.1016/0012-365X(93)E0098-O   DOI
10 B. Wei, Hamiltonian cycles in 1-tough graphs, Graphs Combin. 12 (1996), no. 4, 385-395. https://doi.org/10.1007/BF01858471   DOI
11 R. K. Yeh, A survey on labeling graphs with a condition at distance two, Discrete Math. 306 (2006), no. 12, 1217-1231. https://doi.org/10.1016/j.disc.2005.11.029   DOI
12 J. R. Griggs and R. K. Yeh, Labelling graphs with a condition at distance 2, SIAM J. Discrete Math. 5 (1992), no. 4, 586-595. https://doi.org/10.1137/0405048   DOI
13 F. Maffray and B. A. Reed, A description of claw-free perfect graphs, J. Combin. Theory Ser. B 75 (1999), no. 1, 134-156. https://doi.org/10.1006/jctb.1998.1872   DOI
14 Z. Shao and R. Solis-Oba, On some results for the L(2, 1)-labeling on Cartesian sum graphs, Ars Combin. 124 (2016), 365-377.
15 J. P. Georges, D. W. Mauro, and M. I. Stein, Labeling products of complete graphs with a condition at distance two, SIAM J. Discrete Math. 14 (2001), no. 1, 28-35. https://doi.org/10.1137/S0895480199351859   DOI
16 H. Hajiabolhassan, M. L. Mehrabadi, and R. Tusserkani, Tabular graphs and chromatic sum, Discrete Math. 304 (2005), no. 1-3, 11-22. https://doi.org/10.1016/j.disc.2005.04.022   DOI
17 B. Li, H. J. Broersma, and S. Zhang, Forbidden subgraphs for Hamiltonicity of 1-tough graphs, Discuss. Math. Graph Theory 36 (2016), no. 4, 915-929. https://doi.org/10.7151/dmgt.1897   DOI
18 D. A. Fotakis and P. G. Spirakis, A Hamiltonian approach to the assignment of nonreusable frequencies, in Foundations of software technology and theoretical computer science (Chennai, 1998), 18-29, Lecture Notes in Comput. Sci., 1530, Springer, Berlin, 1998. https://doi.org/10.1007/978-3-540-49382-2_3   DOI
19 Y.-Z. Huang, C. Chiang, L. Huang, and H. Yeh, On L(2, 1)-labeling of generalized Petersen graphs, J. Comb. Optim. 24 (2012), no. 3, 266-279. https://doi.org/10.1007/s10878-011-9380-8   DOI
20 D. Kuo and J.-H. Yan, On L(2, 1)-labelings of Cartesian products of paths and cycles, Discrete Math. 283 (2004), no. 1-3, 137-144. https://doi.org/10.1016/j.disc.2003.11.009   DOI
21 L. Rao, A sufficient condition for 1-tough graphs to be Hamiltonian, in Graph theory, combinatorics, and algorithms, Vol. 1, 2 (Kalamazoo, MI, 1992), 977-980, Wiley-Intersci. Publ, Wiley, New York, 1995.
22 X. Li, B. Wei, Z. Yu, and Y. Zhu, Hamilton cycles in 1-tough triangle-free graphs, Discrete Math. 254 (2002), no. 1-3, 275-287. https://doi.org/10.1016/S0012-365X(01)00358-2   DOI
23 D. Lu, W. Lin, and Z. Song, Distance two labelings of Cartesian products of complete graphs, Ars Combin. 104 (2012), 33-40.
24 C. Lu and Q. Zhou, Path covering number and L(2, 1)-labeling number of graphs, Discrete Appl. Math. 161 (2013), no. 13-14, 2062-2074. https://doi.org/10.1016/j.dam.2013.02.020   DOI
25 J. Fiala, J. Kratochvil, and A. Proskurowski, Distance constrained labeling of precolored trees, in Theoretical computer science (Torino, 2001), 285-292, Lecture Notes in Comput. Sci., 2202, Springer, Berlin, 2001. https://doi.org/10.1007/3-540-45446-2_18   DOI
26 L. Lovasz, J. Nesetril, and A. Pultr, On a product dimension of graphs, J. Combin. Theory Ser. B 29 (1980), no. 1, 47-67. https://doi.org/10.1016/0095-8956(80)90043-X   DOI