62 research outputs found
DP-4-coloring of planar graphs with some restrictions on cycles
DP-coloring was introduced by Dvo\v{r}\'{a}k and Postle as a generalization
of list coloring. It was originally used to solve a longstanding conjecture by
Borodin, stating that every planar graph without cycles of lengths 4 to 8 is
3-choosable. In this paper, we give three sufficient conditions for a planar
graph is DP-4-colorable. Actually all the results (Theorem 1.3, 1.4 and 1.7)
are stated in the "color extendability" form, and uniformly proved by vertex
identification and discharging method.Comment: 13 pages, 5 figures. arXiv admin note: text overlap with
arXiv:1908.0490
Filling the complexity gaps for colouring planar and bounded degree graphs.
We consider a natural restriction of the List Colouring problem, k-Regular List Colouring, which corresponds to the List Colouring problem where every list has size exactly k. We give a complete classification of the complexity of k-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no 4-cycles and no 5-cycles. We also give a complete classification of the complexity of this problem and a number of related colouring problems for graphs with bounded maximum degree
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