Radio k-colorings of paths

• Autorzy: Gary Chartrand , Ladislav Nebeský , Ping Zhang
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

radio k-coloring; radio k-chromatic number

Kategorie tematyczne

• 05C12: Distance in graphs
• 05C15: Coloring of graphs and hypergraphs
• 05C78: Graph labelling (graceful graphs, bandwidth, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2004
• Tom: 24
• Numer: 1
• Strony: 5-21

Daty

wydano

2004

otrzymano

2000-12-16

poprawiono

2002-11-14

Twórcy

autor

Gary Chartrand

• Department of Mathematics, Western Michigan University, Kalamazoo, MI 49008, USA

autor

Ladislav Nebeský

• Faculty of Arts and Philosophy, Charles University, Prague nám. J. Palacha 2, CZ - 116 38 Praha 1, Czech Republic

autor

Ping Zhang

• 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.

Identyfikatory

DOI

10.7151/dmgt.1209