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Graph Theory » Coloring » Vertex coloring

Partial List Coloring ★★★

Author(s): Iradmusa

Let $ G $ be a simple graph, and for every list assignment $ \mathcal{L} $ let $ \lambda_{\mathcal{L}} $ be the maximum number of vertices of $ G $ which are colorable with respect to $ \mathcal{L} $. Define $ \lambda_t = \min{ \lambda_{\mathcal{L}} } $, where the minimum is taken over all list assignments $ \mathcal{L} $ with $ |\mathcal{L}| = t $ for all $ v \in V(G) $.

Conjecture   [2] Let $ G $ be a graph with list chromatic number $ \chi_\ell $ and $ 1\leq r\leq s\leq \chi_\ell $. Then \[\frac{\lambda_r}{r}\geq\frac{\lambda_s}{s}.\]

Keywords: list assignment; list coloring

Posted by Iradmusa
updated May 12th, 2008
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Graph Theory » Coloring » Vertex coloring

Partial List Coloring ★★★

Author(s): Albertson; Grossman; Haas

Conjecture   Let $ G $ be a simple graph with $ n $ vertices and list chromatic number $ \chi_\ell(G) $. Suppose that $ 0\leq t\leq \chi_\ell $ and each vertex of $ G $ is assigned a list of $ t $ colors. Then at least $ \frac{tn}{\chi_\ell(G)} $ vertices of $ G $ can be colored from these lists.

Keywords: list assignment; list coloring

Posted by Iradmusa
updated May 5th, 2008
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