29 research outputs found
Online List Colorings with the Fixed Number of Colors
The online list coloring is a widely studied topic in graph theory. A graph
is 2-paintable if we always have a strategy to complete a coloring in an
online list coloring of in which each vertex has a color list of size 2. In
this paper, we focus on the online list coloring game in which the number of
colors is known in advance. We say that is -paintable if we always
have a strategy to complete a coloring in an online list coloring of in
which we know that there are exactly colors in advance, and each vertex has
a color list of size 2.
Let denote the maximum in which is not -paintable, and
denote the minimum in which is not -paintable. We
show that if is not 2-paintable, then and Furthermore, we characterize with and
respectively
Complexity of equitable tree-coloring problems
A \emph{-tree-coloring} of a graph is a -coloring of vertices
of such that the subgraph induced by each color class is a forest of
maximum degree at most A \emph{-tree-coloring} of a graph
is a -coloring of vertices of such that the subgraph induced by each
color class is a forest.
Wu, Zhang, and Li introduced the concept of \emph{equitable -tree-coloring} (respectively, \emph{equitable -tree-coloring})
which is a -tree-coloring (respectively, -tree-coloring)
such that the sizes of any two color classes differ by at most one. Among other
results, they obtained a sharp upper bound on the minimum such that
has an equitable -tree-coloring for every
In this paper, we obtain a polynomial time criterion to decide if a complete
bipartite graph has an equitable -tree-coloring or an equitable
-tree-coloring. Nevertheless, deciding if a graph in general
has an equitable -tree-coloring or an equitable
-tree-coloring is NP-complete.Comment: arXiv admin note: text overlap with arXiv:1506.0391
Equitable colorings of complete multipartite graphs
A -\emph{equitable coloring} of a graph is a proper -coloring such
that the sizes of any two color classes differ by at most one. In contrast with
ordinary coloring, a graph may have an equitable -coloring but has no
equitable -coloring. The \emph{equitable chromatic threshold} is the
minimum such that has an equitable -coloring for every
In this paper, we establish the notion of which can
be computed in linear-time and prove the following. Assume that
has an equitable -coloring. Then is the minimum such that has an equitable
-coloring for each satisfying Since
has an equitable -coloring, the
equitable chromatic threshold of is
We find out later that the aforementioned immediate consequence is exactly
the same as the formula of Yan and Wang \cite{YW12}. Nonetheless, the notion of
can be used for each in which
has an equitable -coloring and the proof presented here is much shorter.Comment: arXiv admin note: text overlap with arXiv:1506.0391
The strong equitable vertex 2-arboricity of complete bipartite and tripartite graphs
A \emph{-tree-coloring} of a graph is a -coloring of vertices
of such that the subgraph induced by each color class is a forest of
maximum degree at most An \emph{equitable -tree-coloring} of a
graph is a -tree-coloring such that the sizes of any two color
classes differ by at most one. Let the \emph{strong equitable vertex
-arboricity} be the minimum such that has an equitable -tree-coloring for every
In this paper, we find the exact value for each and
$va^\equiv_2(K_{l,m,n}).
Generalization of some results on list coloring and DP-coloring
In this work, we introduce DPG-coloring using the concepts of DP-coloring and
variable degeneracy to modify the proofs on the following papers: (i)
DP-3-coloring of planar graphs without , -cycles and cycles of two
lengths from (R. Liu, S. Loeb, M. Rolek, Y. Yin, G. Yu, Graphs
and Combinatorics 35(3) (2019) 695-705), (ii) Every planar graph without
-cycles adjacent simultaneously to -cycles and -cycles is
DP--colorable when (P. Sittitrai, K. Nakprasit,
arXiv:1801.06760(2019) preprint), (iii) Every planar graph is -choosable (C.
Thomassen, J. Combin. Theory Ser. B 62 (1994) 180-181). Using this
modification, we obtain more results on list coloring, DP-coloring,
list-forested coloring, and variable degeneracy.Comment: arXiv admin note: text overlap with arXiv:1807.0081
Sufficient conditions on cycles that make planar graphs 4-choosable
Xu and Wu proved that if every -cycle of a planar graph is not
simultaneously adjacent to -cycles and -cycles, then is
-choosable. In this paper, we improve this result as follows. Let For any chosen if every -cycle of a planar graph
is not simultaneously adjacent to -cycles, -cycles, and -cycles,
then is -choosable
Planar graphs without pairwise adjacent 3-,4-,5-, and 6-cycle are 4-choosable
Xu and Wu proved that if every 5-cycle of a planar graph G is not
simultaneously adjacent to 3-cycles and 4-cycles, then G is 4-choosable. In
this paper, we improve this result as follows. If G is a planar graph without
pairwise adjacent 3-,4-,5-, and 6-cycle, then G is 4-choosable.Comment: 17 pages and 2 figure
Every planar graph without -cycles adjacent simultaneously to -cycles and -cycles is DP--colorable when
DP-coloring is a generalization of a list coloring in simple graphs. Many
results in list coloring can be generalized in those of DP-coloring. Kim and
Ozeki showed that planar graphs without -cycles where or are
DP--colorable. Recently, Kim and Yu extended the result on - and
-cycles by showing that planar graphs without triangles adjacent to
-cycles are DP--colorable. Xu and Wu showed that planar graphs without
-cycles adjacent simultaneously to -cycles and -cycles are
-choosable. In this paper, we extend the result on -cycles and triangles
adjacent to -cycles by showing that planar graphs without -cycles
adjacent simultaneously to -cycles and -cycles are DP--colorable when
This also generalizes the result of Xu and Wu
Analogue of DP-coloring on variable degeneracy and its applications on list vertex-arboricity and DP-coloring
In \cite{listnoC3adjC4}), Borodin and Ivanova proved that every planar graph
without -cycles adjacent to -cycle is list vertex--aborable. In fact,
they proved a more general result. Inspired by these results and DP-coloring
which becomes a widely studied topic, we introduce a generalization on variable
degeneracy including list vertex arboricity. We use this notion to extend a
general result by Borodin and Ivanova. Not only that this theorem implies
results about planar graphs without -cycles adjacent to -cycle by Borodin
and Ivanova, it implies many other results including a result by Kim and Yu
\cite{KimY} that every planar graph without -cycles adjacent to -cycle is
DP--colorable.Comment: 8 pages, 5 figure
Sufficient conditions on planar graphs to have a relaxed DP--colorability
It is known that DP-coloring is a generalization of a list coloring in simple
graphs and many results in list coloring can be generalized in those of
DP-coloring. In this work, we introduce a relaxed DP-coloring which is a
generalization if a relaxed list coloring. We also shows that every planar
graph without -cycles or -cycles is DP--colorable. It
follows immediately that is -choosable.Comment: arXiv admin note: substantial text overlap with arXiv:1801.0676