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## Discussiones Mathematicae Graph Theory

2007 | 27 | 1 | 105-123
Tytuł artykułu

### Linear and cyclic radio k-labelings of trees

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Motivated by problems in radio channel assignments, we consider radio k-labelings of graphs. For a connected graph G and an integer k ≥ 1, a linear radio k-labeling of G is an assignment f of nonnegative integers to the vertices of G such that
$|f(x)-f(y)| ≥ k+1-d_G(x,y)$,
for any two distinct vertices x and y, where $d_G(x,y)$ is the distance between x and y in G. A cyclic k-labeling of G is defined analogously by using the cyclic metric on the labels. In both cases, we are interested in minimizing the span of the labeling. The linear (cyclic, respectively) radio k-labeling number of G is the minimum span of a linear (cyclic, respectively) radio k-labeling of G. In this paper, linear and cyclic radio k-labeling numbers of paths, stars and trees are studied. For the path Pₙ of order n ≤ k+1, we completely determine the cyclic and linear radio k-labeling numbers. For 1 ≤ k ≤ n-2, a new improved lower bound for the linear radio k-labeling number is presented. Moreover, we give the exact value of the linear radio k-labeling number of stars and we present an upper bound for the linear radio k-labeling number of trees.
Słowa kluczowe
EN
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
105-123
Opis fizyczny
Daty
wydano
2007
otrzymano
2005-12-12
poprawiono
2006-08-26
Twórcy
autor
• LE2I, UMR CNRS 5158, Université de Bourgogne, 21078 Dijon cedex, France
autor
• LE2I, UMR CNRS 5158, Université de Bourgogne, 21078 Dijon cedex, France
autor
• LE2I, UMR CNRS 5158, Université de Bourgogne, 21078 Dijon cedex, France
Bibliografia
• [1] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of cycles, in: Proceedings of the Thirty-first Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 2000) 144 (2000) 129-141.
• [2] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of graphs, Math. Bohem. 127 (2002) 57-69.
• [3] G. Chartrand, D. Erwin, P. Zhang and F. Harary, Radio labelings of graphs, Bull. Inst. Combin. Appl. 33 (2001) 77-85.
• [4] G. Chartrand, L. Nebeský and P. Zhang, Radio k-colorings of paths, Discuss. Math. Graph Theory 24 (2004) 5-21, doi: 10.7151/dmgt.1209.
• [5] G. Chartrand, T. Thomas and P. Zhang, A new look at Hamiltonian walks, Bull. Inst. Combin. Appl. 42 (2004) 37-52.
• [6] G. Chartrand, T. Thomas, P. Zhang and V. Saenpholphat, On the Hamiltonian number of a graph, Congr. Numer. 165 (2003) 51-64.
• [7] J.R. Griggs and R.K. Yeh, Labelling graphs with a condition at distance 2, SIAM J. Disc. Math. 5 (1992) 586-595, doi: 10.1137/0405048.
• [8] J. van den Heuvel, R. Leese and M. Shepherd, Graph labeling and radio channel assignment, J. Graph Theory 29 (1998) 263-283, doi: 10.1002/(SICI)1097-0118(199812)29:4<263::AID-JGT5>3.0.CO;2-V
• [9] M. Kchikech, R. Khennoufa and O. Togni, Radio k-labelings for cartesian products of graphs, in: Proceedings of the 7th International Conference on Graph Theory (ICGT'05), Electronic Notes in Discrete Mathematics 22 (2005) 347-352. Extended version submited to Discrete Mathematics, doi: 10.1016/j.endm.2005.06.078.
• [10] R. Khennoufa and O. Togni, A note on radio antipodal colourings of paths, Math. Bohemica 130 (2005) 277-282.
• [11] D. Král, L.-D. Tong and X. Zhu, Upper Hamiltonian numbers and Hamiltonian spectra of graphs, Australasian J. Combin. 35 (2006) 329-340.
• [12] R.A. Leese and S.D. Noble, Cyclic labellings with constraints at two distances, The Electronic Journal of Combinatorics 11 (2004).
• [13] D. Liu and X. Zhu, Multi-level distance labelings for paths and cycles, SIAM J. Discrete Math. 19 (2005) 610-621, doi: 10.1137/S0895480102417768.
• [14] V. Saenpholphat, F. Okamoto and P. Zhang, Measures of traceability in graphs, Math. Bohemica 131 (2006) 63-84.
• [15] N. Schabanel, Radio Channel Assignment, (PhD Thesis, Merton College Oxford, 1998).
Typ dokumentu
Bibliografia
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