The List-L(2,1)-labeling Problem of Digraphs

• Main Authors: Jyun-Wei Huang, 黃俊瑋
• Other Authors: David Kuo
• Format: Others
• Language: en_US
• Published: 2007
• Online Access: http://ndltd.ncl.edu.tw/handle/z82zjs

碩士 === 國立東華大學 === 應用數學系 === 95 === Given a graph $G$ with $n$ vertices and a function $L:V(G) ightarrow2^{% %TCIMACRO{U{2115} }% %BeginExpansion mathbb{N} %EndExpansion }$, let $A_{L}= igcuplimits_{vin V(G)}L(v),$ we say that $L$ is $(2,1)$ extit{-choosable for }$G$ if there exists a function $c:Im ={L(v_{i}):1leq ileq n} ightarrow A_{L},$ $c(L(v_{i}))=a_{i}$ for all $i,$ $1leq ileq n$, which satisfies the following conditions: (1) $a_{i}in L(v_{i}),$ (2) $|a_{i}-a_{j}|geqslant2$ $ $if $d_{G}(v_{i},v_{j})=1,$ (3) $|a_{i}-a_{j}|geqslant1$ $ $if $d_{G}(v_{i},v_{j})=2.$ ewline In this case, the function $c$ is said to be a $(2,1)$ extit{-choosable function of }$G$ extit{ with respect to }$L.$ If for all the function $L$ with $|L(v_{i})|geqslant k$ for all $v_{i}in V(G)$, there is a $(2,1)$% -choosable function of $G$ with respect to $L,$ then we say that $G$ has a $k $ extit{-list-}$L(2,1)$ extit{-labeling}. The extit{list-}$L(2,1)$% extit{-labeling number of }$G$, denoted by $lambda_{l}(G)$ , is defined by $lambda_{l}(G)=min{k:G~$has~a $k$-list-$L(2,1)$-labeling$}$. We considered the case when the transmitters have direction constraints, that is, the extit{list-}$L(2,1)$ extit{-labeling on digraphs}. Recall that in a digraph $D$ the distance $d_{D}(x,y)$ from vertex $x$ to vertex $y$ is the length of a shortest dipath (directed path) from $x$ to $y$. We then may define list-$L(2,1)$-labeling, $k$-list-$L(2,1)$-labelings and list-$L(2,1)$% -labeling numbers for digraphs in precisely the same way as for graphs. However, to distinguish with the notation for graphs, we use $overrightarrow {lambda_{l}}(D)$ for the list-$L(2,1)$-labeling number of a digraph $D$. In this paper, we study the list-$L(2,1)$-labeling number of digraphs. We give some basic properties for the list-$L(2,1)$-labeling number of digraphs in Section two, and consider the list-$L(2,1)$-labeling number of those digraphs $D$ whose underline graphs are paths, cycles or trees in Section three. And in the last section, we give the exact value of the list-$L(2,1)$% -labeling number of the digraph $overrightarrow{K_{2,n}}$.