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For a connected graph G of diameter d and an integer k with 1 ≤ k ≤ d, a radio k-coloring of G is an assignment c of colors (positive integers) to the vertices of G such that
d(u,v) + |c(u)- c(v)| ≥ 1 + k
for every two distinct vertices u and v of G, where d(u,v) is the distance between u and v. The value rcₖ(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rcₖ(G) of G is the minimum value of rcₖ(c) taken over all radio k-colorings c of G. In this paper, radio k-colorings of paths are studied. For the path Pₙ of order n ≥ 9 and n odd, a new improved bound for $rc_{n- 2}(Pₙ)$ is presented. For n ≥ 4, it is shown that
$rc_{n-3}(Pₙ) ≤ \binom{n-2} {2}$
Upper and lower bounds are also presented for rcₖ(Pₙ) in terms of k when 1 ≤ k ≤ n- 1. The upper bound is shown to be sharp when 1 ≤ k ≤ 4 and n is sufficiently large.
d(u,v) + |c(u)- c(v)| ≥ 1 + k
for every two distinct vertices u and v of G, where d(u,v) is the distance between u and v. The value rcₖ(c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rcₖ(G) of G is the minimum value of rcₖ(c) taken over all radio k-colorings c of G. In this paper, radio k-colorings of paths are studied. For the path Pₙ of order n ≥ 9 and n odd, a new improved bound for $rc_{n- 2}(Pₙ)$ is presented. For n ≥ 4, it is shown that
$rc_{n-3}(Pₙ) ≤ \binom{n-2} {2}$
Upper and lower bounds are also presented for rcₖ(Pₙ) in terms of k when 1 ≤ k ≤ n- 1. The upper bound is shown to be sharp when 1 ≤ k ≤ 4 and n is sufficiently large.
Słowa kluczowe
Kategorie tematyczne
Wydawca
Czasopismo
Rocznik
Tom
Numer
Strony
5-21
Opis fizyczny
Daty
wydano
2004
otrzymano
2000-12-16
poprawiono
2002-11-14
Twórcy
autor
- Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
autor
- Faculty of Arts and Philosophy, Charles University, Prague nám. J. Palacha 2, CZ - 116 38 Praha 1, Czech Republic
autor
- Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA
Bibliografia
- [1] G. Chartrand, D. Erwin, F. Harary and P. Zhang, Radio labelings of graphs, Bull. Inst. Combin. Appl. 33 (2001) 77-85.
- [2] G. Chartrand, D. Erwin and P. Zhang, A graph labeling problem suggested by FM channel restrictions, Bull. Inst. Combin. Appl. (accepted).
- [3] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of cycles, Congr. Numer. 144 (2000) 129-141.
- [4] G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of graphs, Math. Bohem. 127 (2002) 57-69.
- [5] D. Fotakis, G. Pantziou, G. Pentaris and P. Sprirakis, Frequency assignment in mobile and radio networks, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 45 (1999) 73-90.
- [6] W. Hale, Frequency assignment: theory and applications, Proc. IEEE 68 (1980) 1497-1980, doi: 10.1109/PROC.1980.11899.
- [7] J. van den Heuvel, R.A. Leese and M.A. 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
- [8] Minimum distance separation between stations, Code of Federal Regulations, Title 47, sec. 73.207.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1209
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