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Let $S={1,2,\dots,m+n-1}$. An $m\times n$ matrix($\in S^{m\times n}$) is called silver matrix if (a) There is no same numbers in the row or column. (like latin square) (b) {$i$ th row}$\cup${$i$ th column}=S for all $1\leq i\leq min(m,n)$ Is Does silver matrix exist for all $m\neq n$ ? If this conjecture is true, $d(K_m\times K_n, m+n-1) = mn-min(m,n)$ ($m\neq n$) ($d$ is defining number, $\times$ is cartesian product) |
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Generalized silver matrix (related to defining number)Let $S={1,2,\dots,m+n-1}$. An $m\times n$ matrix($\in S^{m\times n}$) is called silver matrix if (a) There is no same numbers in the row or column. (like latin square) (b) {$i$ th row}$\cup${$i$ th column}=S for all $1\leq i\leq min(m,n)$ Is silver matrix exist for all $m\neq n$ ? If this conjecture is true, $d(K_m\times K_n, m+n-1) = mn-min(m,n)$ ($m\neq n$) ($d$ is defining number, $\times$ is cartesian product)
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