On the IC-colorings of complete tripartite graphs

• Main Authors: Chun-Yi Kuo, 郭俊億
• Other Authors: Chin-Lin Shiue
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/49537015849169965661

碩士 === 中原大學 === 應用數學研究所 === 99 === Let $G$ be a graph and let $f$ be a function which maps $V(G)$ into the set of positive integers. We define $f(H)=Sigma_{v in V(H)}f(v)$ for each subgraph $H$ of $G$. We say $f$ to be an extit{IC-coloring} of $G$ if for any integer $k in [1,f(G)]$ there is a connected subgraph $H$ of $G$ such that $f(H)=k$. Clearly, any connected graph $G$ admits an IC-coloring. The extit{IC-index} of a graph $G$, denoted by $M(G)$, is defined to be $M(G)= maxleftlbrace f(G)mid ight.$ $f$ is an IC-coloring of $left. G ight brace$. If $f$ is an IC-coloring of $G$ such that $f(G) = M(G)$, then we say that $f$ is an maximal IC-coloring of $G$. In this thesis, we prove that $M(K_{m_{1},m_{2},m_{3}})= 13cdot2^{m_{1}+m_{2}+m_{3}-4}-3cdot2^{m_{1}-2}+4$ for $2leq m_{1}leq m_{2}leq m_{3}$.