Publication at Faculty of Mathematics and Physics |
2017
Abstract
We answer positively the question of Albertson asking whether every planar graph can be 5-list-colored even if it contains precolored vertices, as long as they are sufficiently far apart from each other. In order to prove this claim, we also give bounds on the sizes of graphs critical with respect to 5-list coloring.
In particular, if G is a planar graph, H is a connected subgraph of G and L is an assignment of lists of colors to the vertices of G such that vertical bar L(v)vertical bar >= 5 for every v is an element of V(G)\V(H) and G is not L-colorable, then G contains a subgraph with O(vertical bar H vertical bar(2)) vertices that is not L-colorable.