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packing
Inequality of the means ★★★
Author(s):
Question Is is possible to pack
rectangular
-dimensional boxes each of which has side lengths
inside an
-dimensional cube with side length
?
![$ n^n $](/files/tex/699fff17a123a14fe73275fc0a636f46688029d9.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ a_1,a_2,\ldots,a_n $](/files/tex/e7587d53923e6a1289b1a25bce813c459b94a973.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ a_1 + a_2 + \ldots a_n $](/files/tex/e1dd2dd4a0225f650d6b7185f7abde703d0fbdbd.png)
Keywords: arithmetic mean; geometric mean; Inequality; packing
Ding's tau_r vs. tau conjecture ★★★
Author(s): Ding
Conjecture Let
be an integer and let
be a minor minimal clutter with
. Then either
has a
minor for some
or
has Lehman's property.
![$ r \ge 2 $](/files/tex/44f910a811a9d2212e22c6bd7fabe7de0bd5e7fe.png)
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
![$ \frac{1}{r}\tau_r(H) < \tau(H) $](/files/tex/b886bbe2e88e54cd298fbc37bbbe43ea9b09c9b7.png)
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
![$ J_k $](/files/tex/ba650479e114120ba9553d070affc34746d309b2.png)
![$ k \ge 2 $](/files/tex/1fb800e3b693294681248f4481b3c041cdcaf1d3.png)
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
Keywords: clutter; covering; MFMC property; packing
Woodall's Conjecture ★★★
Author(s): Woodall
Conjecture If
is a directed graph with smallest directed cut of size
, then
has
disjoint dijoins.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
Ryser's conjecture ★★★
Author(s): Ryser
Conjecture Let
be an
-uniform
-partite hypergraph. If
is the maximum number of pairwise disjoint edges in
, and
is the size of the smallest set of vertices which meets every edge, then
.
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ \nu $](/files/tex/84ef96eb1030eaa1447d7a3c9ce8be567b7eccc8.png)
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
![$ \tau $](/files/tex/706065ef4c2d4b462dff91b9ad9fc69d846c15dc.png)
![$ \tau \le (r-1) \nu $](/files/tex/03ec477958b1e8cdfcb12ec5ef3d32014472a61d.png)
Keywords: hypergraph; matching; packing
Packing T-joins ★★
Author(s): DeVos
Conjecture There exists a fixed constant
(probably
suffices) so that every graft with minimum
-cut size at least
contains a
-join packing of size at least
.
![$ c $](/files/tex/dccee841f3f498c2c58fa6ae1c1403c5a88c5b8d.png)
![$ c=1 $](/files/tex/7a8b9609d823e7ffb81159e3dfcbd5cb2599e11d.png)
![$ T $](/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png)
![$ k $](/files/tex/c450c3185f7285cfa0b88d3a903c54f7df601201.png)
![$ T $](/files/tex/79f55d2e1d83a7726c807a70cbe756713b0437b6.png)
![$ (2/3)k-c $](/files/tex/da50fbb1472ce7fd6fa00876af3022e1be50d2ec.png)
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