A note on 3-choosability of planar graphs without certain cycles

AbstractSteinberg asked whether every planar graph without 4 and 5 cycles is 3-colorable. Borodin, and independently Sanders and Zhao, showed that every planar graph without any cycle of length between 4 and 9 is 3-colorable. We improve this result by showing that every planar graph without any cycle of length 4, 5, 6, or 9 is 3-choosable

Disproof of the List Hadwiger Conjecture

The List Hadwiger Conjecture asserts that every $K_t$-minor-free graph is $t$-choosable. We disprove this conjecture by constructing a $K_{3t+2}$-minor-free graph that is not $4t$-choosable for every integer $t\geq 1$

On edge-group choosability of graphs

In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our claim are graphs with maximum degree less than 4, planar graphs with maximum degree at least 11, planar graphs without small cycles, outerplanar graphs and near-outerplanar graphs