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list assignment
Partial List Coloring ★★★
Author(s): Iradmusa
Let be a simple graph, and for every list assignment
let
be the maximum number of vertices of
which are colorable with respect to
. Define
, where the minimum is taken over all list assignments
with
for all
.
Conjecture [2] Let
be a graph with list chromatic number
and
. Then
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \chi_\ell $](/files/tex/46e2d5b6cf87b7e24d9ac72345043107b7ccae3a.png)
![$ 1\leq r\leq s\leq \chi_\ell $](/files/tex/a3fc6cd4c98fabb35869474a493492a0435792e7.png)
![\[\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.\]](/files/tex/47be18e956355dd433b88b66eabf01a9e3ed5f61.png)
Keywords: list assignment; list coloring
Partial List Coloring ★★★
Author(s): Albertson; Grossman; Haas
Conjecture Let
be a simple graph with
vertices and list chromatic number
. Suppose that
and each vertex of
is assigned a list of
colors. Then at least
vertices of
can be colored from these lists.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ n $](/files/tex/ec63d7020a64c039d5f6703b8fa3ab7393358b5b.png)
![$ \chi_\ell(G) $](/files/tex/b68082745a25a09294e2c92c006b61d3ef1a9e54.png)
![$ 0\leq t\leq \chi_\ell $](/files/tex/1aee119babfe25de435c7cf6beff3114a5ae9326.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ t $](/files/tex/4761b031c89840e8cd2cda5b53fbc90c308530f3.png)
![$ \frac{tn}{\chi_\ell(G)} $](/files/tex/59b72b19d6799e1fd7a0fc093bff9283068dc838.png)
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
Keywords: list assignment; list coloring
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