51 research outputs found
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
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
Extending List Colorings of Planar Graphs
In the study of list colorings of graphs, we assume each vertex of a graph has a specified list of colors from which it may be colored. For planar graphs, it is known that there is a coloring for any list assignment where each list contains five colors. If we have some vertices that are precolored, can we extend this to a coloring of the entire graph? We explore distance constraints when we allow the lists to contain an extra color. For lists of length five, we fix as a subset of such that all vertices in have been assigned colors from their respective lists. We give a new, simplified proof where there are a small number of precolored vertices on the same face. We also explore cases where and has a separating or between and
DP-3-coloring of planar graphs without certain cycles
DP-coloring is a generalization of list coloring, which was introduced by
Dvo\v{r}\'{a}k and Postle [J. Combin. Theory Ser. B 129 (2018) 38--54]. Zhang
[Inform. Process. Lett. 113 (9) (2013) 354--356] showed that every planar graph
with neither adjacent triangles nor 5-, 6-, 9-cycles is 3-choosable. Liu et al.
[Discrete Math. 342 (2019) 178--189] showed that every planar graph without 4-,
5-, 6- and 9-cycles is DP-3-colorable. In this paper, we show that every planar
graph with neither adjacent triangles nor 5-, 6-, 9-cycles is DP-3-colorable,
which generalizes these results. Yu et al. gave three Bordeaux-type results by
showing that (i) every planar graph with the distance of triangles at least
three and no 4-, 5-cycles is DP-3-colorable; (ii) every planar graph with the
distance of triangles at least two and no 4-, 5-, 6-cycles is DP-3-colorable;
(iii) every planar graph with the distance of triangles at least two and no 5-,
6-, 7-cycles is DP-3-colorable. We also give two Bordeaux-type results in the
last section: (i) every plane graph with neither 5-, 6-, 8-cycles nor triangles
at distance less than two is DP-3-colorable; (ii) every plane graph with
neither 4-, 5-, 7-cycles nor triangles at distance less than two is
DP-3-colorable.Comment: 16 pages, 4 figure
On List Coloring with Separation of the Complete Graph and Set System Intersections
We consider the following list coloring with separation problem: Given a
graph and integers , find the largest integer such that for any
list assignment of with for any vertex and for any edge of , there exists an assignment of
sets of integers to the vertices of such that and
for any vertex and
for any edge . Such a value of is called the separation number of
. Using a special partition of a set of lists for which we obtain an
improved version of Poincar\'e's crible, we determine the separation number of
the complete graph for some values of and , and prove bounds for
the remaining values.Comment: 18 page
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