![](/files/happy5.png)
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)
As you see this conjecture in the special case , is the conjecture of Albertson, Grossman and Haas [1]:
for any
.
Bibliography
[1] M. Albertson, S. Grossman and R. Haas, Partial list colouring, Discrete Math., 214(2000), pp. 235-240.
[2] Moharram N. Iradmusa, A Note on Partial List Colorings, Australasian Journal of Combinatorics, Vol.46, 2010, .
* indicates original appearance(s) of problem.