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\newtheorem{pretheorem}{{\bf ì‰Ì‰ƒ‰‚ }}
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\newtheorem{prelemma}{{\bf ó‰İ }}
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\newtheorem{preexample}{{\bf õ‰·‰‘ñ }}
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\newtheorem{preprob}{{\bf õ‰Æ‰\hamze ‘ó‰‚ }}
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\newenvironment{prop}{\ \\[3mm]{\bf —‰ã‰Âş‰Ó.} \rm }{\\[-5mm]}
\newenvironment{proof}{\ \\[3mm]{\bf �™‰±‰‘–.} \rm }{\hfill{\InE{}$\Box$\EnE{}} \\[1mm]}
\newenvironment{solution}{\ \\[3mm]{\bf Ÿ‰Û.} \rm }{\hfill{\InE{}$\Box$\EnE{}} \\[1mm]}

\title{Ÿ‰\kasre À •‰‘ş‰ƒ‰€‰ü “‰Â�ı —‰ã‰À�\kasre ¢ ¤÷‰Ú‰ƒ‰ß‰‘ö\footnote{\InE{}Transversal\EnE{}}û‰‘\kasre ı õ‰€‰µ‰Ñ‰İ ¢¤ ¤÷‰Ù�õ‰ƒ‰Ã\kasre ı ş‰‘ó‰\kasre ü ğ‰Â�\kasre é \InE{}$K_n$\EnE{}}
\author{“‰ú‰À�¢ �¨‰Ô‰ú‰±‰À\quad 䉱‰‘¢�ó‰Ü‰‚ õ‰½‰Ş‰�¢ş‰‘ö}
\date{}
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\maketitle
\begin{abstract}
õ‰Â“‰âû‰‘ı „—‰ƒ‰ß ø õ‰±‰‘Ÿ‰\kasre ¶ õ‰Â—‰±‰Í “‰‘ �öû‰‘ ş‰Ø‰ü �¥ ›‰‘ó‰°—‰Âş‰ß ø •‰Â¤“‰Â¢—‰Âş‰ß “‰¿‰Çû‰‘ı
—‰Â‰±‰ƒ‰‘– õ‰ü“‰‘ª‰€‰À, ¢¤ �ş‰ß õ‰Ö‰‘ó‰‚ “‰‘ —‰ã‰‘¤ş‰\kasre Ó ¥õ‰‘ö“‰€‰À\kasre ı —‰�¤÷‰Ş‰€‰\kasre ´ ¨‰‘䉵‰ü ø
¤÷‰Ú‰ƒ‰ß‰‘ö �ª‰€‰‘ õ‰üª‰�ş‰İ, ¨‰³‰Å ¤øª‰ü “‰Â�ı ş‰‘ꉵ‰\kasre ß �‰€‰À ¤÷‰Ú‰ƒ‰ß‰‘ö ¢¤ ğ‰Â�é
\InE{}$K_n$\EnE{} �¤�ş‰‚ õ‰ü¢û‰ƒ‰İ.
\end{abstract}
\section{¥õ‰‘ö“‰€‰À\kasre ı —‰�¤÷‰Ş‰€‰\kasre ´ ¨‰‘䉵‰ü ø ¤÷‰Ú‰ƒ‰ß‰‘ö}
ş‰× ¥õ‰‘ö“‰€‰Àı “‰Â�ı ş‰× —‰�¤÷‰Ş‰€‰´ “‰Â�“‰Â �¨‰´ “‰‘ �ê‰Â�\kasre ¥ ş‰‘ñû‰‘ı \InE{}$K_n$\EnE{} “‰‚ \InE{}$n-1$\EnE{} —‰Î‰‘“‰\kasre Õ
õ‰Û. ì‰Ì‰ƒ‰‚ı õ‰È‰ú‰�¤ı ø›‰�¢ ¢�¤¢  “‰Â�ı û‰Â \InE{}$n>4$\EnE{} ¥øš, ş‰‘ñû‰‘ı \InE{}$K_n$\EnE{} 쉑“‰\kasre Û ¤÷‰Ù�õ‰ƒ‰Ãı
“‰‘ \InE{}$n-1$\EnE{} ¤÷‰Ù �¨‰´. ¤ø\kasre © ¨‰‘¡‰\kasre ´ ş‰× ÷‰Ş‰�÷‰‚ �¥ �ş‰ß ¤÷‰Ù�õ‰ƒ‰Ãı)¥õ‰‘ö“‰€‰Àı( ¢¤ \kasre ó‰\kasre İ
¥ş‰Â �õ‰Àù �¨‰´:
\begin{lemma}
“‰‚ �¥�ı û‰Â \InE{}$n>4$\EnE{} ¥øš, ş‰‘ñû‰‘ı \InE{}$K_n$\EnE{} 쉑“‰\kasre Û ¤÷‰Ù�õ‰ƒ‰Ãı “‰‘ \InE{}$n-1$\EnE{} ¤÷‰Ù �¨‰´.
\end{lemma}
\begin{proof}
¤�§û‰‘ı \InE{}$K_n$\EnE{} ¤� “‰‘ 䉀‰‘¬‰\kasre  õ‰¹‰Ş‰�䉂ı \InE{}$\{\infty,0,1,2,\dots,n-2\}$\EnE{} ª‰Ş‰‘¤ùğ‰Á�¤ı õ‰ü‰ƒ‰İ.
ş‰× ¤÷‰Ù�õ‰ƒ‰Ãı �¥ \InE{}$K_n$\EnE{} “‰‚ �ş‰ß ¬‰�¤– õ‰ã‰Âê‰ü õ‰ü‰ƒ‰İ: ş‰‘ñû‰‘ı “‰‘
¤÷‰\kasre Ù \InE{}$i$\EnE{} )\InE{}$0\leq i\leq n-2$\EnE{}(, 䉱‰‘¤—‰€‰À �¥
$$\{\{\infty,i\},\{i-1,i+1\},\{i-2,i+2\},\ldots,
\{i-\frac{n}{2}+1,i+\frac{n}{2}-1\}\}\pmod{n-1}$$
“‰‚ �ş‰ß ¤÷‰Ù�õ‰ƒ‰Ãı, ¤÷‰Ù�õ‰ƒ‰Ã\kasre ı ¨‰‘䉵‰ü õ‰üğ‰�ş‰ƒ‰İ ø �ğ‰Â ¤�\kasre § \InE{}$\infty$\EnE{}
¤� õ‰Âî‰Ã ş‰× ¢�ş‰Âù ì‰Â�¤ ¢û‰ƒ‰İ ø ¤�§û‰‘ı 0 —‰‘ \InE{}$n-2$\EnE{} ¤� “‰‘ ê‰��¬‰\kasre Û “‰Â�“‰Â ¤øı õ‰½‰ƒ‰\kasre Í
¢�ş‰Âù “‰Ú‰Á�¤ş‰İ, ş‰‘ñû‰‘ı “‰‘ ¤÷‰Ù \InE{}$i$\EnE{} “‰Â�“‰Â÷‰À “‰‘ ş‰‘ó‰ü  õ‰Âî‰Ã ¤� “‰‚ ¤�\kasre § \InE{}$i$\EnE{}
ø¬‰Û õ‰ü‰À ø î‰Ü‰ƒ‰‚ ş‰‘ñû‰‘ı 䉪‰�¢ “‰Â �ö. ¢¤ ÷‰µ‰ƒ‰¹‰‚ �‰�ö \InE{}$n-1$\EnE{} ê‰Â¢ �¨‰´ ¢ø ş‰‘ñ ¢¤
¬‰�¤—‰ü û‰Ş‰Â÷‰Ù ¡‰��û‰€‰À “‰�¢ :
\begin{enumerate}
\item “‰Â û‰İ 䉪‰�¢ “‰‘ª‰€‰À,  ¢¤�ş‰ß ¬‰�¤– ş‰Ø‰ü �¥ �öû‰‘ ş‰‘\kasre ñ õ‰Âî‰Ãı)ş‰× ¨‰Â �ö \InE{}$\infty$\EnE{}
�¨‰´( ø ¢ş‰Ú‰Âı ş‰‘\kasre ñ 艃‰Âõ‰Âî‰Ãı �¨‰´.
\item “‰‘ û‰İ õ‰��¥ı “‰‘ª‰€‰À,  ¢¤ �ş‰ß ¬‰�¤– û‰Â ¢ø ş‰‘\kasre ñ 艃‰Âõ‰Âî‰Ãı û‰Æ‰µ‰€‰À.
\end{enumerate}
û‰Ş‰»‰€‰ƒ‰ß û‰ƒ‰º ¢ø ş‰‘\kasre ñ õ‰Âî‰Ãı û‰Ş‰Â÷‰Ù ÷‰ƒ‰Æ‰µ‰€‰À ø �ğ‰Â ω�\kasre ñ ş‰× ş‰‘\kasre ñ 艃‰Âõ‰Âî‰Ãı ¤� “‰Â�“‰Â
1 “‰‚�®‰‘ꉂı Ÿ‰À�ì‰\kasre Û —‰ã‰À�¢ ¤�‰�\kasre § ¤øı õ‰Æ‰ƒ‰Âı “‰Ú‰ƒ‰Âş‰İ  ¤øı õ‰½‰ƒ‰\kasre Í ¢�ş‰Âù ş‰× ¨‰\kasre  ş‰‘ñ ¤�
“‰‚ ¨‰\kasre  ¢ş‰Ú‰Â ø¬‰Û õ‰ü‰À ø ω�\kasre ñ ş‰‘ñû‰‘ı õ‰Âî‰Ãı ¤� ¬‰Ô‰Â ¢¤ ÷‰Ñ‰Â “‰Ú‰ƒ‰Âş‰İ, �÷‰Ú‰‘ù û‰Â ¤÷‰Ù
�¥ ş‰‘ñû‰‘ı “‰‚ ω�ñ 0 —‰‘ \InE{}$\frac{n}{2}-1$\EnE{}, �¥ û‰Â î‰À�ô ¢ì‰ƒ‰Ö‰\nasb ‘ ş‰× ä‰Ì‰� ¢�¤¢, ø ¢¤
÷‰µ‰ƒ‰¹‰‚ û‰ƒ‰º ¢ø ş‰‘\kasre ñ “‰‘ ω�\kasre ñ ş‰×¨‰‘ö û‰Ş‰Â÷‰Ù ÷‰ƒ‰Æ‰µ‰€‰À.
\english
\begin{center}
\input{pic1.pic}
\end{center}
\farsi
\end{proof}
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ş‰× ¤÷‰Ú‰ƒ‰ß‰‘ö ¢¤ ş‰× ¤÷‰Ù�õ‰ƒ‰Ãı ş‰‘ó‰ü ş‰× ğ‰Â�é, —‰Î‰‘“‰\kasre Õ î‰‘õ‰Ü‰ü �¨‰´  û‰ƒ‰º ¢ø ş‰‘ó‰ü
¢¤ �ö ¤÷‰\kasre Ù ş‰×¨‰‘÷‰ü ÷‰À�¤÷‰À. Ÿ‰À�‰Â —‰ã‰À�¢ õ‰Ş‰Ø‰ß ¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı ¢ø“‰‚¢ø ş‰‘ñõ‰¹‰Ã�
¢¤ ğ‰Â�é \InE{}$K_n$\EnE{}, \InE{}$n-1$\EnE{} �¨‰´  ş‰‘ñû‰‘ı ğ‰Â�é ¤� �ê‰Â�¥ õ‰ü‰À. û‰Ş‰»‰€‰ƒ‰ß ™‰‘“‰´ ª‰Àù
�¨‰´  û‰Â ¤÷‰Ù�õ‰ƒ‰Ãı ş‰‘ó‰ü \InE{}$K_n$\EnE{} Ÿ‰À�ì‰Û ş‰× ¤÷‰Ú‰ƒ‰ß‰‘ö ¢�¤¢. ø›‰�\kasre ¢ �‰€‰À ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö
ş‰‘ñõ‰¹‰Ã� ¢¤ ¤÷‰Ù�õ‰ƒ‰Ãı ¨‰‘䉵‰ü õ‰�®‰�á “‰½‰\kasre ¶ õ‰‘ �¨‰´. ¢¤ ø�ì‰â û‰€‰�¥ õ‰È‰¿‰É ÷‰È‰Àù �¨‰´ 
\InE{}$K_n$\EnE{} �‰€‰À ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö ¢ø“‰‚¢ø ş‰‘ñõ‰¹‰Ã� ¢�¤¢ ø ş‰‘ “‰Â�ı �‰‚ \InE{}$n$\EnE{}û‰‘ş‰ü \InE{}$K_n$\EnE{} 쉑“‰Û
�ê‰Â�¥ “‰‚ ¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı ¢ø“‰‚¢ø ş‰‘ñõ‰¹‰Ã� �¨‰´.
%
\section{¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı õ‰€‰µ‰Ñ‰İ ¢¤ ¤÷‰Ù �õ‰ƒ‰Ãı ¨‰‘䉵‰ü \InE{}$K_n$\EnE{}}
ş‰× ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö õ‰€‰µ‰Ñ‰İ ¤� �‰€‰ƒ‰ß —‰ã‰Âş‰Ó õ‰ü‰ƒ‰İ: ¤÷‰Ú‰ƒ‰ß‰‘÷‰ü  �¥ ş‰× ş‰‘ñ “‰‚
ω�\kasre ñ 0 ø \InE{}$\frac{n}{2}-1$\EnE{} ş‰‘ñ “‰‘ ω�\kasre ñ “‰Â�“‰Â —‰È‰Ø‰ƒ‰Û ª‰Àù “‰‘ª‰À.
%
“‰‚ ÷‰Ñ‰Â õ‰ü¤¨‰À ø›‰�¢ ÷‰À�ª‰µ‰‚ “‰‘ª‰À \InE{}$n>2$\EnE{}�ı  \InE{}$K_n$\EnE{} 쉑“‰\kasre Û �ê‰Â�¥ “‰‚ �‰€‰À ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö
ş‰‘ñõ‰¹‰Ã� “‰‘ª‰À øó‰ü ¢¤ õ‰�¤\kasre ¢ ¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı õ‰€‰µ‰Ñ‰İ �ş‰ß õ‰�®‰�á “‰‚ ¤�Ÿ‰µ‰ü �™‰±‰‘– õ‰üª‰�¢:
\begin{theorem}
“‰‚ �¥�ı û‰ƒ‰º \InE{}$n>2$\EnE{}�ı, \InE{}$K_n$\EnE{} 쉑“‰\kasre Û �ê‰Â�¥ “‰‚ ¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı õ‰€‰µ‰Ñ‰İ ÷‰ƒ‰Æ‰´.
\end{theorem}
\begin{proof}
õ‰ü¢�÷‰ƒ‰İ û‰Â ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö õ‰€‰µ‰Ñ‰İ \InE{}$\frac{n}{2}-1$\EnE{} ş‰‘ñ “‰‘ ω�\kasre ñ “‰Â�“‰Â)\InE{}$k$\EnE{}( ¢�¤¢ ø
î‰Ü‰ƒ‰‚ı ş‰‘ñû‰‘ı “‰‚ ω�\kasre ñ \InE{}$k$\EnE{} û‰İ “‰‘ş‰À “‰‚¤ ¤ø÷‰À, øó‰ü —‰ã‰À�¢ ş‰‘ñû‰‘ı “‰‚ ω�ñ \InE{}$k$\EnE{}, \InE{}$n-1$\EnE{}
�¨‰´  “‰Â \InE{}$\frac{n}{2}-1$\EnE{} “‰¿‰Ç•‰Áş‰Â ÷‰ƒ‰Æ‰´, ¢¤ ø�ì‰â ¢¤ “‰ú‰µ‰Âş‰ß Ÿ‰‘ó‰´)“‰ƒ‰È‰µ‰Âş‰ß —‰ã‰À�\kasre ¢
¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı õ‰€‰µ‰Ñ‰\kasre İ ş‰‘ñõ‰¹‰Ã�ı õ‰Ş‰Ø‰ß( õ‰ü—‰��÷‰ƒ‰İ \InE{}$n-2$\EnE{} ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö õ‰€‰µ‰Ñ‰\kasre İ ş‰‘ñõ‰¹‰Ã�
¢�ª‰µ‰‚ “‰‘ª‰ƒ‰İ  ¢¤ �ş‰ß ¬‰�¤– “‰‚ �¥�ı û‰Â \InE{}$0\leq k\leq \frac{n}{2}-1$\EnE{} ¢ì‰ƒ‰Ö‰\nasb ‘ ş‰×
ş‰‘ñ “‰‚ ω�\kasre ñ \InE{}$k$\EnE{} ¢¤ û‰ƒ‰º ¤÷‰Ú‰ƒ‰ß‰‘÷‰ü ª‰Â ÷‰À�¤¢. �ş‰ß —‰ã‰À�¢)\InE{}$n-2$\EnE{}(
¤÷‰Ú‰ƒ‰ß‰‘\kasre ö õ‰€‰µ‰Ñ‰\kasre İ ş‰‘ñõ‰¹‰Ã� “‰Â�ı \InE{}$n=6$\EnE{} ø›‰�¢ ¢�¤¢.
\end{proof}
%
\begin{prop}
\InE{}$T_{n,k}$\EnE{} )\InE{}$n$\EnE{} ¥øš ø \InE{}$2k|n-2$\EnE{}( “‰Â�“‰Â �¨‰´ “‰‘ —‰Î‰‘“‰Õ õ‰Ü‰ü ¢¤ \InE{}$K_n$\EnE{}  �¥ ş‰‘\kasre ñ \InE{}$\{0,\infty\}$\EnE{} ø
\InE{}$\frac{n}{2}-1$\EnE{} ş‰‘ñ “‰‚ ω�\kasre ñ \InE{}$k$\EnE{} —‰È‰Ø‰ƒ‰Û ª‰Àù ø “‰‚ ª‰Ø‰\kasre Û ¥ş‰Â �¨‰´:
$$
\{\{0,\infty\}\}+\{\{i+2jk,i+(2j+1)k\}|1\leq i\leq k,i+(2j+1)k\leq n-2,0
\leq j\leq \frac{n-2}{2k}\}
$$
\end{prop}
%
\begin{theorem}
�ğ‰Â \InE{}$n=4m$\EnE{}, \InE{}$T_{n,k}$\EnE{} û‰ƒ‰ºğ‰‘ù ş‰× ¤÷‰Ú‰ƒ‰ß‰‘ö ÷‰ƒ‰Æ‰´ øó‰ü �ğ‰Â \InE{}$n=4m+2$\EnE{},
\InE{}$T_{n,k}$\EnE{} û‰Ş‰��¤ù ¤÷‰Ú‰ƒ‰ß‰‘ö �¨‰´.
\end{theorem}
\begin{proof}
¢¤ ì‰Æ‰Ş‰\kasre ´ ş‰\kasre × ª‰Ø‰\kasre Û ¥ş‰Â õ‰ã‰€‰‘ı ¢¨‰µ‰‚ ¢¤ \InE{}$T_{n,k}$\EnE{} ÷‰È‰‘ö ¢�¢ù ª‰Àù �¨‰´. ê‰Â­ õ‰ü ‰ƒ‰İ —‰ã‰À�¢
¢¨‰µ‰‚û‰‘ \InE{}$\alpha$\EnE{} “‰‘ª‰À •‰Å \InE{}$\alpha=\frac{n-2}{2k}$\EnE{} ø ¢¤ û‰Â ¢¨‰µ‰‚ \InE{}$2k$\EnE{} ¤�§ ø›‰�¢ ¢�¤¢.
�¥ �÷‰¹‰‘  ş‰‘ñû‰‘ı “‰‚ ω�\kasre ñ \InE{}$k$\EnE{} ÷‰Ş‰ü—‰��÷‰€‰À û‰Ş‰Â÷‰Ù “‰‘ª‰€‰À •‰Å \InE{}$T_{n,k}$\EnE{} ¤÷‰Ú‰ƒ‰ß‰‘ö
�¨‰´ �ğ‰Â ø ê‰Ö‰Í �ğ‰Â ş‰‘\kasre ñ \InE{}$\{0,\infty\}$\EnE{} “‰‘ û‰ƒ‰º î‰À�ô �¥ ş‰‘ñû‰‘ı “‰‚ ω�\kasre ñ \InE{}$k$\EnE{} û‰Ş‰Â÷‰Ù ÷‰±‰‘ª‰À.
�‰�ö û‰ƒ‰º ş‰‘ó‰ü “‰ƒ‰ß ¤�‰�§ ¢ø ¢¨‰µ‰‚ ø›‰�¢ ÷‰À�¤¢ ş‰‘ñû‰‘ı ¢¤øö û‰Â ¢¨‰µ‰‚ ¤� “‰Â¤¨‰ü
õ‰ü‰ƒ‰İ. “‰‚ ¤�Ÿ‰µ‰ü õ‰ü—‰��ö ¢ş‰À —‰€‰ú‰‘ ø쉵‰ü ş‰‘ó‰ü 䉪‰�¢ “‰Â ş‰‘ñ \InE{}$\{0,\infty\}$\EnE{} ø›‰�¢
¢�¤¢  \InE{}$\alpha$\EnE{} ø \InE{}$k$\EnE{} û‰Â ¢ø ê‰Â¢ “‰‘ª‰€‰À �‰�ö �ğ‰Â \InE{}$\alpha$\EnE{} ¥øš “‰‘ª‰À ş‰‘ó‰ü “‰ƒ‰ß
¤�‰�§ 1 —‰‘ \InE{}$\frac{n-2}{2}$\EnE{} ø \InE{}$\frac{n-2}{2}+1$\EnE{} —‰‘ \InE{}$n-2$\EnE{} ø›‰�¢ ÷‰À�¤¢ ø �ğ‰Â
\InE{}$\alpha$\EnE{} ê‰Â¢ ø \InE{}$k$\EnE{} ¥øš “‰‘ª‰À û‰İ ş‰‘ó‰ü õ‰��¥ı \InE{}$\{0,\infty\}$\EnE{} ÷‰Ş‰üª‰�¢.
\begin{enumerate}
\item \InE{}$n=4m$\EnE{}: ¢¤ �ş‰ß ¬‰�¤– \InE{}$2k\alpha = n-2 = 4m-2$\EnE{} •‰Å \InE{}$k\alpha = 2m-1$\EnE{},
¢¤ ÷‰µ‰ƒ‰¹‰‚ \InE{}$k$\EnE{} ø \InE{}$\alpha$\EnE{} ê‰Â¢ û‰Æ‰µ‰€‰À •‰Å ş‰‘ó‰ü û‰Ş‰Â÷‰Ù “‰‘ \InE{}$\{0,\infty\}$\EnE{} ø›‰�¢ ¢�¤¢
ø ÷‰µ‰ƒ‰¹‰‚ õ‰üª‰�¢ \InE{}$T_{n,k}$\EnE{} ş‰× ¤÷‰Ú‰ƒ‰ß‰‘ö ÷‰ƒ‰Æ‰´.
\item \InE{}$n=4m+2$\EnE{}: ¢¤ �ş‰ß ¬‰�¤– \InE{}$2k\alpha = n-2 = 4m$\EnE{} •‰Å \InE{}$k\alpha = 2m$\EnE{},
¢¤ ÷‰µ‰ƒ‰¹‰‚ Ÿ‰À�ì‰Û ş‰Ø‰ü �¥ \InE{}$k$\EnE{} ø \InE{}$\alpha$\EnE{} ¥øš �¨‰´ •‰Å
ş‰‘ó‰ü û‰Ş‰Â÷‰Ù “‰‘ \InE{}$\{0,\infty\}$\EnE{} ø›‰�¢ ÷‰À�¤¢ ø ÷‰µ‰ƒ‰¹‰‚ õ‰üª‰�¢ \InE{}$T_{n,k}$\EnE{} ş‰× ¤÷‰Ú‰ƒ‰ß‰‘ö �¨‰´.
\end{enumerate}
\english
\begin{center}
\input{pic24.pic}
\input{pic23.pic}
\input{pic22.pic}
\input{pic21.pic}
\end{center}
\farsi
\end{proof}
%
\vspace{-10mm}
\begin{prop}
\InE{}$T_{n,k}^{(i)}$\EnE{}: û‰Ş‰‘ö \InE{}$T_{n,k}$\EnE{} �¨‰´  �ä‰À�¢ �ö û‰Ş‰Ú‰ü “‰‚ û‰€‰Ù \InE{}$(n-1)$\EnE{} “‰‘ \InE{}$i$\EnE{} ›‰Ş‰â ª‰Àù�÷‰À.
“‰‚ ä‰Ü‰´ —‰Ö‰‘¤ö õ‰ü—‰��ö ¢ş‰À  “‰‚ �¥�ı û‰Â \InE{}$i\in Z$\EnE{}, \InE{}$T_{n,k}^{(i)}$\EnE{} ¤÷‰Ú‰ƒ‰ß‰‘ö �¨‰´ �ğ‰Â ø
ê‰Ö‰Í �ğ‰Â \InE{}$T_{n,k}$\EnE{} ¤÷‰Ú‰ƒ‰ß‰‘ö “‰‘ª‰À.
\end{prop}
%
\begin{lemma}
\InE{}$T_{n,k}^{(i)}$\EnE{} ø \InE{}$T_{n,k'}^{(i')}$\EnE{} ¢ø ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö ş‰‘ñõ‰¹‰Ã� û‰Æ‰µ‰€‰À �ğ‰Â ø
ê‰Ö‰Í �ğ‰Â û‰Âî‰À�ô “‰‚ —‰€‰ú‰‘ş‰ü ş‰× ¤÷‰Ú‰ƒ‰ß‰‘ö “‰�¢ù ø
\begin{enumerate}
\item \InE{}$k\not = k'$\EnE{} �÷‰Ú‰‘ù \InE{}$i\not=i'$\EnE{}
\item \InE{}$k=k'$\EnE{} �÷‰Ú‰‘ù \InE{}$|i-i'|\equiv k \pmod{n-1}$\EnE{}
\end{enumerate}
\end{lemma}
\begin{proof}
�ğ‰Â \InE{}$k\not = k'$\EnE{} ¢¤ ÷‰µ‰ƒ‰¹‰‚ —‰€‰ú‰‘ ş‰‘ñû‰‘ı û‰İω�\kasre ñ ¢ø ¤÷‰Ú‰ƒ‰ß‰‘ö ş‰‘ñû‰‘ı \InE{}$\{\infty,i\}$\EnE{} ø
\InE{}$\{\infty,i'\}$\EnE{} û‰Æ‰µ‰€‰À  �ğ‰Â \InE{}$i\not=i'$\EnE{}, �ş‰ß ¢ø ÷‰ƒ‰Ã õ‰¹‰Ã� ¡‰��û‰€‰À “‰�¢. øó‰ü �ğ‰Â
\InE{}$k=k'$\EnE{}, �ğ‰Â \InE{}$|i-i'|\not\equiv k \pmod{n-1}$\EnE{} �öğ‰‘ù ¢ø ¤÷‰Ú‰ƒ‰ß‰‘ö Ÿ‰À�ì‰Û ¢¤ ş‰×
ş‰‘ñ “‰‚ ω�\kasre ñ \InE{}$k$\EnE{} �ª‰µ‰Â�í ¡‰��û‰€‰À ¢�ª‰´ øó‰ü ¢¤ 艃‰Â �ş‰ß¬‰�¤– ω±‰Õ —‰ã‰Âş‰\kasre Ó \InE{}$T_{n,k}$\EnE{}
õ‰ü—‰��ö ¢ş‰À  �ş‰ß ¢ø, ş‰‘ñõ‰¹‰Ã� ¡‰��û‰€‰À “‰�¢.
\end{proof}
%
\begin{theorem}
“‰‚ �¥�ı û‰Â \InE{}$n=4m+2$\EnE{}  \InE{}$n>10$\EnE{}, \InE{}$K_n$\EnE{} Ÿ‰À�ì‰Û 4 “‰‚䉅øùı —‰ã‰À�¢ õ‰Ö‰Æ‰�ôä‰Ü‰ƒ‰‚û‰‘ı
\InE{}$\frac{n-2}{2}$\EnE{} ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö õ‰€‰µ‰Ñ‰\kasre İ ş‰‘ñõ‰¹‰Ã� ¢�¤¢.
\end{theorem}
\begin{proof}
\InE{}$T_{n,1}^{(0)}$\EnE{}, \InE{}$T_{n,1}^{(1)}$\EnE{}, \InE{}$T_{n,2}^{(2)}$\EnE{}, \InE{}$T_{n,2}^{(4)}$\EnE{},
\InE{}$T_{n,\frac{n-2}{2}}^{(3)}$\EnE{}, \InE{}$T_{n,\frac{n-2}{2}}^{(3+\frac{n-2}{2})}$\EnE{},
\InE{}$T_{n,\frac{n-2}{4}}^{(5)}$\EnE{} ø \InE{}$T_{n,\frac{n-2}{4}}^{(5+\frac{n-2}{4})}$\EnE{}
—‰È‰Ø‰ƒ‰\kasre Û 8 ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö ş‰‘ñõ‰¹‰Ã� õ‰ü¢û‰€‰À. Ÿ‰‘ñ “‰‚ �¥�ı û‰Â õ‰Ö‰Æ‰�ôä‰Ü‰ƒ‰\kasre ‚ \InE{}$\frac{n-2}{2}$\EnE{}
õ‰·‰\kasre Û \InE{}$k$\EnE{}, 艃‰Â �¥ 1, 2, \InE{}$\frac{n-2}{4}$\EnE{} ø \InE{}$\frac{n-2}{2}$\EnE{}, ş‰× \InE{}$i$\EnE{} ¤�  ş‰‘\kasre ñ
\InE{}$\{\infty,i\}$\EnE{} �¨‰µ‰Ô‰‘¢ù ÷‰È‰Àù •‰ƒ‰À� î‰Â¢ù)�‰€‰ƒ‰ß \InE{}$i$\EnE{}�ı Ÿ‰µ‰Ş‰‘ ø›‰�¢ ¢�¤¢ ¥ş‰Â� �ğ‰Â
ø›‰�¢ ÷‰À�ª‰µ‰‚ “‰‘ª‰À “‰‚ �ş‰ß õ‰ã‰€‰ü �¨‰´  \InE{}$n-1$\EnE{} ¤÷‰Ú‰ƒ‰ß‰‘\kasre ö ş‰‘ñõ‰¹‰Ã� �÷‰µ‰¿‰‘’ î‰Â¢ù�ş‰İ(
ø \InE{}$T_{n,K}^{(i)}$\EnE{} ¤� “‰‚ ¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ş‰Ş‰‘ö õ‰ü�ê‰Ã�ş‰ƒ‰İ, �ş‰ß —‰ã‰À�¢, û‰Ş‰‘ö —‰ã‰À�\kasre ¢
¡‰��¨‰µ‰‚ ª‰Àù �¨‰´.
\end{proof}
%
\begin{center}
\bf ›‰Àø\kasre ñ Ÿ‰À�‰\kasre  —‰ã‰À�\kasre ¢ ¤÷‰Ú‰ƒ‰ß‰‘öû‰‘ı ¢ø“‰‚¢ø ş‰‘ñõ‰¹‰Ã�ı \\
ğ‰Â�é \InE{}$K_n$\EnE{} )\InE{}$t_n$\EnE{}( “‰Â�ı \InE{}$n$\EnE{}û‰‘ı î‰��‰× \\ \vspace{5mm}
\begin{tabular}{|c|c|}
\hline
\InE{}$n$\EnE{} & \InE{}$t_n$\EnE{} \\
\hline
2  & 1 \\
4  & 0 \\
6  & 4 \\
8  & 3 \\
01 & \InE{}$\geq  8$\EnE{} \\
21 & \InE{}$\geq 10$\EnE{} \\
41 & \InE{}$\geq 11$\EnE{} \\
61 & \InE{}$\geq 12$\EnE{} \\
81 & \InE{}$\geq 13$\EnE{} \\
02 & \InE{}$\geq 14$\EnE{} \\
22 & \InE{}$\geq 15$\EnE{} \\
42 & \InE{}$\geq 15$\EnE{} \\
62 & \InE{}$\geq 15$\EnE{} \\
\hline
\end{tabular}
\end{center}
%
\begin{thebibliography}{99}
%
\bibitem{} ¢î‰µ‰Â ¨‰\tashdid ƒ‰À 䉱‰‘¢�ó‰‚ õ‰½‰Ş‰�¢ş‰‘ö ø õ‰ú‰€‰À§ õ‰½‰\tashdid މÀ õ‰ú‰Àş‰‘ö,
{\khabide ¤ş‰È‰‚û‰‘ı ş‰× õ‰Æ‰‘ó‰‚ı �󉪉³‰ƒ‰‘\kasre ¢ ›‰ú‰‘÷‰ü ¤ş‰‘®‰ü ¨‰‘ñ 7991}, õ‰¹‰Ü‰‚ı ¤ª‰\kasre À �õ‰�¥\kasre ©
¤ş‰‘®‰ü, ¨‰‘\kasre ñ ¢ø�¥¢û‰İ, ª‰Ş‰‘¤ùı 15, “‰ú‰‘\kasre ¤ 7731 ø ê‰Ê‰Ü‰€‰‘õ‰‚ı ¢�÷‰Ç•‰Äøù, ª‰Ş‰‘¤ùı 2,
¥õ‰Æ‰µ‰‘\kasre ö 7731.
%
\english
\bibitem{m1} Charles J. Colbourn and Jeffrey H. Dinitz,
{\em Latin Squares, MOLs and Orthogonal arrays}, 
The CRC Handbook of Combinatorial Designs.
\farsi
%
\end{thebibliography}
%
\end{document}