53 research outputs found
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
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
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
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}).
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
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
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
Some Inequalities for the Polar Derivative of Some Classes of Polynomials
In this paper, we investigate an upper bound of the polar derivative of a
polynomial of degree
where zeros are in and the remaining
zeros are outside where
Furthermore, we give a lower bound of this polynomial where zeros
are outside and the remaining
zeros are in where Comment: 17 page
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
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