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Albertson, Michael O.
Crossing numbers and coloring ★★★
Author(s): Albertson
We let denote the crossing number of a graph
.
Conjecture Every graph
with
satisfies
.
![$ G $](/files/tex/b8e7ad0330f925492bf468b5c379baec88cf1b3d.png)
![$ \chi(G) \ge t $](/files/tex/cb74f630a3b502fe0bd0726e72880c005ec02d22.png)
![$ cr(G) \ge cr(K_t) $](/files/tex/64703b245f27999c87b0a8e3c56c14e80f1925d4.png)
Keywords: coloring; complete graph; crossing number
Partial List Coloring ★★★
Author(s): Albertson; Grossman; Haas
Conjecture Let
be a simple graph with
vertices and list chromatic number
. Suppose that
and each vertex of
is assigned a list of
colors. Then at least
vertices of
can be colored from these lists.
![$ 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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