Importance: High ✭✭✭
Author(s): Ding, Guoli
Recomm. for undergrads: no
Posted by: mdevos
on: November 11th, 2007

\begin{conjecture} Let $r \ge 2$ be an integer and let $H$ be a minor minimal \Def{clutter} with $\frac{1}{r}\tau_r(H) < \tau(H)$. Then either $H$ has a $J_k$ minor for some $k \ge 2$ or $H$ has Lehman's property. \end{conjecture}

See \Def[Wikipedia's Clutter]{clutter (mathematics)} for definitions of clutter and clutter minors. The clutter $J_k$ is the degenerate projective plane with vertex set $\{0,1,\ldots,k\}$ and edge set $\{ \{0,1\}, \{1,2\},\ldots,\{0,k\},\{0,1,\ldots,k\} \}$. If $H=(V,E)$ is a clutter, then for every positive integer $r$ we let $\tau_r(H)$ denote the largest multiset of vertices of $H$ which hit every edge at least $r$ times. Note that $\tau(H) = \tau_1(H)$ and that $\tau_r(H) \le r \tau(H)$.

We say that a clutter $H$ with $|V(H)| = n$, $\tau(H) = s$ and $\tau(b(H)) = r$ has \emph{Lehman's property} if $rs > n$, $E(H) = \{A_1,\ldots,A_n\}$, $E(b(H)) = \{B_1,\ldots,B_n\}$, and the following properties are satisfied. \begin{itemize} \item $|A_i| = r$ for every $1 \le i \le n$. \item $|B_i| = s$ for every $1 \le i \le n$. \item $|A_i \cap B_i| = rs - n +1$ for $1 \le i \le n$ \item $|A_i \cap B_j| = 1$ if $1 \le i,j \le n$ and $i \neq j$. \item every $v \in V(H)$ lies in exactly $r$ edges of $H$, $s$ edges of $b(H)$, and $rs-n+1$ members of $\{A_1 \cap B_1, \ldots ,A_n \cap B_n\}$. \end{itemize}

Although the conditions in Lehman's condition are extremely stringent, Lehman [L] showed that every minor minimal clutter with the MFMC property satisfies these properties. Since the MFMC property for $H$ implies $\frac{1}{r}\tau_r(H) = \tau(H)$ (and the degenerate projective planes are minor minimal without MFMC), if true, the above conjecture would be a nice extension of Lehman's theorem.

Ding [D] proved this conjecture for $r=2$, but it is open for all other cases.

Bibliography

*[D] G. Ding, Clutters with tau_2=2 tau, Discrete Math. 115 (1993), no. 1-3, 141--152. \MRhref{MR1217624}.

[L] A. Lehman, On the width-length inequality, mimeographic notes, published 1979. Math. Program. 17, 403--417 \MRhref{MR0550854}.


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