158 research outputs found
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
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
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}).
Equitable vertex arboricity of graphs
An equitable -tree-coloring of a graph is a coloring to vertices
of such that the sizes of any two color classes differ by at most one and
the subgraph induced by each color class is a forest of maximum degree at most
and diameter at most . The minimum such that has an equitable
-tree-coloring for every is called the strong equitable
-vertex-arboricity and denoted by . In this paper,
we give sharp upper bounds for and
by showing that
and
va^{\equiv}_{k,\infty}(K_{n,n})=O(n^{\1/2}) for every . It is also
proved that for every planar graph
with girth at least 5 and for every
planar graph with girth at least 6 and for every outerplanar graph. We
conjecture that for every planar graph
and for
every graph
A Note on the Equitable Choosability of Complete Bipartite Graphs
In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of
equitable coloring called equitable choosability. A -assignment, , for a
graph assigns a list, , of available colors to each ,
and an equitable -coloring of is a proper coloring, , of such
that for each and each color class of has size
at most . Graph is said to be equitably
-choosable if an equitable -coloring of exists whenever is a
-assignment for . In this note we study the equitable choosability of
complete bipartite graphs. A result of Kostochka, Pelsmajer, and West implies
is equitably -choosable if provided . We prove is equitably -choosable if which gives is equitably
-choosable for certain satisfying . We also give a
complete characterization of the equitable choosability of complete bipartite
graphs that have a partite set of size at most 2.Comment: 9 page
A Simple Characterization of Proportionally 2-choosable Graphs
We recently introduced proportional choosability, a new list analogue of
equitable coloring. Like equitable coloring, and unlike list equitable coloring
(a.k.a. equitable choosability), proportional choosability bounds sizes of
color classes both from above and from below. In this note, we show that a
graph is proportionally 2-choosable if and only if it is a linear forest such
that its largest component has at most 5 vertices and all of its other
components have two or fewer vertices. We also construct examples that show
that characterizing equitably 2-choosable graphs is still open.Comment: 9 page
Equitable Coloring and Equitable Choosability of Planar Graphs without chordal 4- and 6-Cycles
A graph is equitably -choosable if, for any given -uniform list
assignment , is -colorable and each color appears on at most
vertices. A graph is equitably -colorable if
the vertex set can be partitioned into independent subsets ,
, , such that for .
In this paper, we prove that if is a planar graph without chordal - and
-cycles, then is equitably -colorable and equitably -choosable
where .Comment: 21 pages,3 figure
Total Colourings - A survey
The smallest integer needed for the assignment of colors to the elements
so that the coloring is proper (vertices and edges) is called the total
chromatic number of a graph. Vizing and Behzed conjectured that the total
coloring can be done using at most colors, where is
the maximum degree of .
It is not settled even for planar graphs. In this paper we give a survey on
total coloring of graphs.Comment: 23 pages, 3 figures 1 tabl
On List Equitable Total Colorings of the Generalized Theta Graph
In 2003 Kostochka, Pelsmajer, and West introduced a list analogue of
equitable coloring called equitable choosability. A -assignment, , for a
graph assigns a list, , of available colors to each ,
and an equitable -coloring of is a proper coloring, , of such
that for each and each color class of has size
at most . In 2018, Kaul, Mudrock, and Pelsmajer
subsequently introduced the List Equitable Total Coloring Conjecture which
states that if is a total graph of some simple graph, then is equitably
-choosable for each where
is the maximum degree of a vertex in and is the
list chromatic number of . In this paper we verify the List Equitable Total
Coloring Conjecture for subdivisions of stars and the generalized theta graph.Comment: 16 page
Problem collection from the IML programme: Graphs, Hypergraphs, and Computing
This collection of problems and conjectures is based on a subset of the open
problems from the seminar series and the problem sessions of the Institut
Mitag-Leffler programme Graphs, Hypergraphs, and Computing. Each problem
contributor has provided a write up of their proposed problem and the
collection has been edited by Klas Markstr\"om.Comment: This problem collection is published as part of the IML preprint
series for the research programme and also available there
http://www.mittag-leffler.se/research-programs/preprint-series?course_id=4401.
arXiv admin note: text overlap with arXiv:1403.5975, arXiv:0706.4101 by other
author
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