24 research outputs found
Some results on (a:b)-choosability
A solution to a problem of Erd\H{o}s, Rubin and Taylor is obtained by showing
that if a graph is -choosable, and , then is not
necessarily -choosable. Applying probabilistic methods, an upper bound
for the choice number of a graph is given. We also prove that a
directed graph with maximum outdegree and no odd directed cycle is
-choosable for every . Other results presented in this
article are related to the strong choice number of graphs (a generalization of
the strong chromatic number). We conclude with complexity analysis of some
decision problems related to graph choosability
On Two problems of defective choosability
Given positive integers , and a non-negative integer , we say a
graph is -choosable if for every list assignment with
for each and ,
there exists an -coloring of such that each monochromatic subgraph has
maximum degree at most . In particular, -choosable means
-colorable, -choosable means -choosable and
-choosable means -defective -choosable. This paper proves
that there are 1-defective 3-choosable graphs that are not 4-choosable, and for
any positive integers , and non-negative integer , there
are -choosable graphs that are not -choosable.
These results answer questions asked by Wang and Xu [SIAM J. Discrete Math. 27,
4(2013), 2020-2037], and Kang [J. Graph Theory 73, 3(2013), 342-353],
respectively. Our construction of -choosable but not -choosable graphs generalizes the construction of Kr\'{a}l' and Sgall
in [J. Graph Theory 49, 3(2005), 177-186] for the case .Comment: 12 pages, 4 figure
On Structure of Some Plane Graphs with Application to Choosability
AbstractA graph G=(V, E) is (x, y)-choosable for integers x>y⩾1 if for any given family {A(v)∣v∈V} of sets A(v) of cardinality x, there exists a collection {B(v)∣v∈V} of subsets B(v)⊂A(v) of cardinality y such that B(u)∩B(v)=∅ whenever uv∈E(G). In this paper, structures of some plane graphs, including plane graphs with minimum degree 4, are studied. Using these results, we may show that if G is free of k-cycles for some k∈{3, 4, 5, 6}, or if any two triangles in G have distance at least 2, then G is (4m, m)-choosable for all nonnegative integers m. When m=1, (4m, m)-choosable is simply 4-choosable. So these conditions are also sufficient for a plane graph to be 4-choosable
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