3,223 research outputs found
The linear arboricity conjecture for graphs of low degeneracy
A -linear coloring of a graph is an edge coloring of with
colors so that each color class forms a linear forest -- a forest whose each
connected component is a path. The linear arboricity of is the
minimum integer such that there exists a -linear coloring of .
Akiyama, Exoo and Harary conjectured in 1980 that for every graph ,
where
is the maximum degree of . First, we prove the conjecture for
3-degenerate graphs. This establishes the conjecture for graphs of treewidth at
most 3 and provides an alternative proof for the conjecture in some classes of
graphs like cubic graphs and triangle-free planar graphs for which the
conjecture was already known to be true. Next, for every 2-degenerate graph
, we show that if
. We conjecture that this equality holds also when
and show that this is the case for some well-known
subclasses of 2-degenerate graphs. All our proofs can be converted into linear
time algorithms.Comment: 23 pages, 6 figures, preliminary version appeared in the proceedings
of WG 202
On vertex coloring without monochromatic triangles
We study a certain relaxation of the classic vertex coloring problem, namely,
a coloring of vertices of undirected, simple graphs, such that there are no
monochromatic triangles. We give the first classification of the problem in
terms of classic and parametrized algorithms. Several computational complexity
results are also presented, which improve on the previous results found in the
literature. We propose the new structural parameter for undirected, simple
graphs -- the triangle-free chromatic number . We bound by
other known structural parameters. We also present two classes of graphs with
interesting coloring properties, that play pivotal role in proving useful
observation about our problem. We give/ask several conjectures/questions
throughout this paper to encourage new research in the area of graph coloring.Comment: Extended abstrac
Planar graph coloring avoiding monochromatic subgraphs: trees and paths make things difficult
We consider the problem of coloring a planar graph with the minimum number of colors such that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem
3-coloring triangle-free planar graphs with a precolored 8-cycle
Let G be a planar triangle-free graph and let C be a cycle in G of length at
most 8. We characterize all situations where a 3-coloring of C does not extend
to a proper 3-coloring of the whole graph.Comment: 20 pages, 5 figure
- …