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Discussiones Mathematicae Graph Theory 24(1) (2004) 5-21
DOI: https://doi.org/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 rc_k (c) of a radio k-coloring c of G is the maximum color assigned to a vertex of G. The radio k-chromatic number rc_k (G) of G is the minimum value of rc_k (c) taken over all radio k-colorings c of G. In this paper, radio k-colorings of paths are studied. For the path P_n of order n ≥ 9 and n odd, a new improved bound for rc_{n− 2} (P_n) is presented. For n ≥ 4, it is shown that

Upper and lower bounds are also presented for rc_k (P_n) 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

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Received 16 December 2000
Revised 14 November 2002