S-Packing Coloring on Graphs

• Main Authors: Sheng-Hua Chen, 陳聖華
• Other Authors: Gerard Jennhwa Chang
• Format: Others
• Language: en_US
• Published: 2009
• Online Access: http://ndltd.ncl.edu.tw/handle/88821028664876324897

碩士 === 國立臺灣大學 === 數學研究所 === 97 === The concept of the $S$-emph{packing coloring} motivates from the areas of frequency or channel assignment in wireless networks, resource placements and biological diversity. For instance, Federal Communications Commission of the United States Government has established numerous rules and regulations concerning the assignment of broadcast frequencies to radio stations. Two radio stations assigned the same broadcast frequency must be located sufficiently far apart so that the broadcast does not interfere with the reception of the other,and because of physical limitation, different frequency require different distance. The $S$-emph{packing coloring} problem is defined as follows: given a finite nondecreasing sequence $(s_1,s_2,dots,s_k)$ of positive integers, a graph $G=(V,E)$ is called $(s_1,s_2,dots,s_k)$-emph{packing colorable} if there is a function $f:V ightarrow {1,2,dots,k}$ such that $d(x,y)>s_i$ or $x=y$ when $f(x)=f(y)=i$. For an infinite non-decreasing sequence $S={s_n}_{n=1}^{infty}$ of positive integers,the $S$-emph{chromatic number} $chi_S(G)$ of $G$ is the minimum number $k$ such that $G$ is $(s_1,s_2,dots,s_k)$-packing colorable. In this thesis, we find some sharp bounds of $S$-chromatic numbers of some classes of graphs. We also characterize graphs which attain the bounds. From a complexity point of view, we distinguish NP-completeness or P-solvablility for some $S$-packing coloring problems.