DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 24(1) (2004) 5-21
DOI: 10.7151/dmgt.1209

RADIO k-COLORINGS OF PATHS

 Gary Chartrand

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

Ladislav Nebeský

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

Ping Zhang

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

Abstract

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 rcn− 2 (Pn) is presented. For n ≥ 4, it is shown that

rcn− 3 (Pn) ≤ ( 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.

Keywords: radio k-coloring, radio k-chromatic number.

2000 Mathematics Subject Classification: 05C12, 05C15, 05C78.

References

[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).
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[8] Minimum distance separation between stations, Code of Federal Regulations, Title 47, sec. 73.207.

Received 16 December 2000
Revised 14 November 2002