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Ding, Guoli
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 $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
Keywords: clutter; covering; MFMC property; packing
Bounded colorings for planar graphs ★★
Author(s): Alon; Ding; Oporowski; Vertigan
Question Does there exists a fixed function
so that every planar graph of maximum degree
has a partition of its vertex set into at most three sets
so that for
, every component of the graph induced by
has size at most
?
![$ f : {\mathbb N} \rightarrow {\mathbb N} $](/files/tex/e5839c90f2b5ca6fe2f58de668c9549b3ad831bd.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ \{V_1,V_2,V_3\} $](/files/tex/4743659cb13aa2fd42df6291dc9839397a771197.png)
![$ i=1,2,3 $](/files/tex/a551cf18cf10c10840b4155cdb12c330c8fec96b.png)
![$ V_i $](/files/tex/af854be1f03aac481e0a165c3908976d4b5b0aa0.png)
![$ f(d) $](/files/tex/eb1c96d175a846e74b707abbc2eabf3ea4a2d7b2.png)
Keywords: coloring; partition; planar graph
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