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Next: $B$3$NJ8=q$K$D$$$F(B... (PDF ե: planar.pdf)

$BJ?@.(B 13 $BG/(B 6 $B7n(B 21 $BF|(B
$K_{3,3}$ $B$H(B $K_5$ $B$,J?LL%0%i%U$G$J$$$3$H$N>ZL@(B
$B?73c9)2JBg3X(B $B>pJsEE;R9)3X2J(B $BC]LnLP<#(B

$B652J=q(B p.96 $BNc(B 3.6 $B$K(B $K_5$ $B$,J?LL%0%i%U$G$J$$$3$H$N>ZL@$,:\$C$F$$$k$,!"(B $B0l2U=j@bL@ITB-$NE@$,$"$k(B ( $\vert E\vert\geq 7\times 3/2$)$B!#$=$NOC$r4^$a$F(B $K_{3,3}$ $B$H(B $K_5$ $B$,J?LL%0%i%U$G$J$$$3$H$N>ZL@$r9T$&!#(B

$B0J2, $\vert E\vert$, $\vert V\vert$ $BEy$N5-9f$N0UL#$K$D$$$F$O652J=q;2>H$N$3$H!#(B


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$BJ?LL%0%i%U$,C1=c%0%i%U$G$"$k$H$-!"(B$R\geq 2$ $B$J$i$P(B

\begin{displaymath}
3R\leq 2\vert E\vert
\end{displaymath}


$B>ZL@(B

$B8r:9$9$kJU$N$J$$C1=c%0%i%U$G$O!"30ItNN0h$b4^$aA4$F$NNN0h$,(B 3 $BK\0J>e$NJU$K0O$^$l$F$$$k!#(B

$B$=$7$F!"3FJU$O(B 2 $B$D$NNN0h$rJ,$1$F$$$k$+$i!"3FNN0h$r0O$`JU$r(B $BA4$F?t$(>e$2$k$H3FJU$r(B 2 $BEY?t$($k$3$H$K$J$k$N$G(B

\begin{displaymath}
\mbox{$B3FNN0h$r0O$`JU$N?t$N9g7W(B} = \mbox{($BJU$N?t(B)}\times 2
\geq 3\times\mbox{($BNN0h$N?t(B)}
\end{displaymath}

$B$H$J$k!#(B



$BJdBj(B 2

$B;03Q7A$,4^$^$l$J$$C1=cJ?LL%0%i%U$G$O!"(B$R\geq 2$ $B$KBP$7$F(B

\begin{displaymath}
4R\leq 2\vert E\vert
\end{displaymath}


$B>ZL@(B

$BJdBj(B 1 $B$N>ZL@$NO@K!$K$h$jL@$i$+(B ($BA4$F$NNN0h$,(B 4 $BK\0J>e$N(B $BJU$K0O$^$l$k(B)$B!#(B


$BNc$($P!"J?LL(B 2 $BIt%0%i%U$K$O;03Q7A$O$J$$(B ($B4q?t3Q7A$O$J$$(B) $B$N$G!"$3$N(B $BITEy<0$,@.$jN)$D!#(B


$B7O(B 3

$B40A4(B 2 $BIt%0%i%U(B $K_{3,3}$ $B$OJ?LL%0%i%U$G$O$J$$!#(B


$B>ZL@(B

$K_{3,3}$ $B$O(B $\vert E\vert=9$, $\vert V\vert=6$, $k=1$ $B$J$N$G!"$b$7J?LL%0%i%U$N7A$K(B $B=q$1$?$H$9$k$H!"%*%$%i!<$N8x<0$K$h$j(B

\begin{displaymath}
R=k+1-\vert V\vert+\vert E\vert=1+1-6+9=5
\end{displaymath}

$B$G$J$1$l$P$J$i$J$$$,!"JdBj(B 2 $B$K$h$j(B

\begin{displaymath}
4R\leq 2\vert E\vert
\end{displaymath}

$B$G$J$i$J$1$l$P$J$i$:!"$=$l$O(B $4R=20$, $2\vert E\vert=18$ $B$KL7=b$9$k!#(B



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Next: $B$3$NJ8=q$K$D$$$F(B...
Shigeharu TAKENO
2001$BG/(B 8$B7n(B 9$BF|(B