Discussiones Mathematicae Graph Theory 27(1) (2007) 159-174
doi: 10.7151/dmgt.1352

[BIBTex] [PDF] [PS]


Yu-Fa Shena, Guo-Ping Zhenga, Wen-Jie Heb

aDepartment of Mathematics and Physics
Hebei Normal University of Science and Technology
Qinhuangdao 066004, P.R. China
bApplied Mathematics Institute
Hebei University of Technology
Tianjin 300130, P.R. China
e-mail: syf030514@163.com (Yu-Fa Shen).


For paths Pn, G. Chartrand, L. Nebeský and P. Zhang showed that ac′(Pn) ≤ (n-2)(n-3)/2+2 for every positive integer n, where ac′(Pn) denotes the nearly antipodal chromatic number of Pn. In this paper we show that ac′(Pn) ≤ (n-2)(n-3)/2−[n/2]− ⎣10/n⎦+7 if n is even positive integer and n ≥ 10, and ac′(Pn) ≤ (n-2)(n-3)/2−[(n−1)/2] −⎣13/n⎦+8 if n is odd positive integer and n ≥ 13. For all even positive integers n ≥ 10 and all odd positive integers n ≥ 13, these results improve the upper bounds for nearly antipodal chromatic number of Pn.

Keywords: radio colorings, nearly antipodal chromatic number, paths.

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


[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. 43 (2005) 43-57.
[3]G. Chartrand, D. Erwin and P. Zhang, Radio antipodal colorings of graphs, Math. Bohem. 127 (2002) 57-69.
[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]D. Fotakis, G. Pantziou, G. Pentaris and P. Spirakis, Frequency assignment in mobile and radio networks, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 45 (1999) 73-90.
[6]R. Khennoufa and O. Togni, A note on radio antipodal colorings of paths, Math. Bohem. 130 (2005) 277-282.
[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.

Received 21 February 2006
Revised 31 October 2006