Summary: | 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 $overrightarrow{K_{2,n}}$.
|