![](/files/happy5.png)
hypergraph
A generalization of Vizing's Theorem? ★★
Author(s): Rosenfeld
Conjecture Let
be a simple
-uniform hypergraph, and assume that every set of
points is contained in at most
edges. Then there exists an
-edge-coloring so that any two edges which share
vertices have distinct colors.
![$ H $](/files/tex/76c7b422c8e228780f70a4f31614cfcf3f831c65.png)
![$ d $](/files/tex/aeba4a4076fc495e8b5df04d874f2911a838883a.png)
![$ d-1 $](/files/tex/377f809b25769f204a72e5d4765cddd8aaabe392.png)
![$ r $](/files/tex/535dee6c3b72bcc4d571239ed00be162ee1e6fbe.png)
![$ r+d-1 $](/files/tex/43f812f49b2b3aee5d87139eaff0e0fe02c47dc8.png)
![$ d-1 $](/files/tex/377f809b25769f204a72e5d4765cddd8aaabe392.png)
Keywords: edge-coloring; hypergraph; Vizing
Ryser's conjecture ★★★
Author(s): Ryser
Conjecture Let
be an
-uniform
-partite hypergraph. If
is the maximum number of pairwise disjoint edges in
, and
is the size of the smallest set of vertices which meets every edge, then
.
![$ 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
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