160 research outputs found
Multi-part Nordhaus-Gaddum type problems for tree-width, Colin de Verdi\`ere type parameters, and Hadwiger number
A traditional Nordhaus-Gaddum problem for a graph parameter is to
find a (tight) upper or lower bound on the sum or product of and
(where denotes the complement of ). An
-decomposition of the complete graph is a partition of
the edges of among spanning subgraphs . A traditional
Nordhaus-Gaddum problem can be viewed as the special case for of a more
general -part sum or product Nordhaus-Gaddum type problem. We determine the
values of the -part sum and product upper bounds asymptotically as goes
to infinity for the parameters tree-width and its variants largeur
d'arborescence, path-width, and proper path-width. We also establish ranges for
the lower bounds for these parameters, and ranges for the upper and lower
bounds of the -part Nordhaus-Gaddum type problems for the parameters
Hadwiger number, the Colin de Verdi\`ere number that is used to
characterize planarity, and its variants and
Nordhaus-Gaddum-type theorem for conflict-free connection number of graphs
An edge-colored graph is \emph{conflict-free connected} if, between each
pair of distinct vertices, there exists a path containing a color used on
exactly one of its edges. The \emph{conflict-free connection number} of a
connected graph , denoted by , is defined as the smallest number of
colors that are needed in order to make conflict-free connected. In this
paper, we determine all trees of order for which , where
and . Then we prove that for a
connected graph , and characterize the graphs with
, respectively. Finally, we get the
Nordhaus-Gaddum-type theorem for the conflict-free connection number of graphs,
and prove that if and are connected, then and , and moreover, or
if and only if one of and
is a tree with maximum degree or a , and the lower
bounds are sharp.Comment: 25 page
A Nordhaus-Gaddum conjecture for the minimum number of distinct eigenvalues of a graph
We propose a Nordhaus-Gaddum conjecture for , the minimum number of
distinct eigenvalues of a symmetric matrix corresponding to a graph : for
every graph excluding four exceptions, we conjecture that , where is the complement of . We compute for all trees
and all graphs with , and hence we verify the conjecture for
trees, unicyclic graphs, graphs with , and for graphs with
Some extremal results on the colorful monochromatic vertex-connectivity of a graph
A path in a vertex-colored graph is called a \emph{vertex-monochromatic path}
if its internal vertices have the same color. A vertex-coloring of a graph is a
\emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for
short), if there is a vertex-monochromatic path joining any two vertices in the
graph. For a connected graph , the \emph{monochromatic vertex-connection
number}, denoted by , is defined to be the maximum number of colors
used in an \emph{MVC-coloring} of . These concepts of vertex-version are
natural generalizations of the colorful monochromatic connectivity of
edge-version, introduced by Caro and Yuster. In this paper, we mainly
investigate the Erd\H{o}s-Gallai-type problems for the monochromatic
vertex-connection number and completely determine the exact value.
Moreover, the Nordhaus-Gaddum-type inequality for is also given.Comment: 15 page
The (vertex-)monochromatic index of a graph
A tree in an edge-colored graph is called a \emph{monochromatic tree}
if all the edges of have the same color. For , a
\emph{monochromatic -tree} in is a monochromatic tree of containing
the vertices of . For a connected graph and a given integer with
, the \emph{-monochromatic index } of is
the maximum number of colors needed such that for each subset
of vertices, there exists a monochromatic -tree. In this paper, we prove
that for any connected graph , for each such
that .
A tree in a vertex-colored graph is called a
\emph{vertex-monochromatic tree} if all the internal vertices of have the
same color. For , a \emph{vertex-monochromatic -tree} in
is a vertex-monochromatic tree of containing the vertices of . For a
connected graph and a given integer with , the
\emph{-monochromatic vertex-index } of is the maximum number
of colors needed such that for each subset of vertices,
there exists a vertex-monochromatic -tree. We show that for a given a
connected graph , and a positive integer with , to decide
whether is NP-complete for each integer such that . We also obtain some Nordhaus-Gaddum-type results for the
-monochromatic vertex-index.Comment: 13 page
More on rainbow disconnection in graphs
Let be a nontrivial edge-colored connected graph. An edge-cut of
is called a rainbow cut if no two edges of it are colored the same. An
edge-colored graph is rainbow disconnected if for every two vertices
and , there exists a rainbow cut. For a connected graph , the
rainbow disconnection number of , denoted by , is defined as the
smallest number of colors that are needed in order to make rainbow
disconnected. In this paper, we first solve a conjecture that determines the
maximum size of a connected graph of order with for given
integers and with , where is odd, posed by
Chartrand et al. in \cite{CDHHZ}. Secondly, we discuss bounds of the rainbow
disconnection numbers for complete multipartite graphs, critical graphs,
minimal graphs with respect to chromatic index and regular graphs, and give the
rainbow disconnection numbers for several special graphs. Finally, we get the
Nordhaus-Gaddum-type theorem for the rainbow disconnection number of graphs. We
prove that if and are both connected, then and . Furthermore, examples are given to show that the upper bounds are
sharp for , and the lower bounds are sharp when .Comment: 14 page
Packing parameters in graphs: New bounds and a solution to an open problem
In this paper, we investigate the packing parameters in graphs. By applying
the Mantel's theorem, We give upper bounds on packing and open packing numbers
of triangle-free graphs along with characterizing the graphs for which the
equalities hold and exhibit sharp Nordhaus-Gaddum type inequalities for packing
numbers. We also solve the open problem of characterizing all connected graphs
with posed in [S. Hamid and S. Saravanakumar, {\em
Packing parameters in graphs}, Discuss Math. Graph Theory, 35 (2015), 5--16]
On the regular k-independence number of graphs
The \emph{regular independence number}, introduced by Albertson and Boutin in
1990, is the maximum cardinality of an independent set of in which all
vertices have equal degree in . Recently, Caro, Hansberg and Pepper
introduced the concept of regular -independence number, which is a natural
generalization of the regular independence number. A \emph{-independent set}
is a set of vertices whose induced subgraph has maximum degree at most . The
\emph{regular -independence number} of , denoted by ,
is defined as the maximum cardinality of a -independent set of in which
all vertices have equal degree in . In this paper, the exact values of the
regular -independence numbers of some special graphs are obtained. We also
get some lower and upper bounds for the regular -independence number of
trees with given diameter, and the lower bounds for the regular
-independence number of line graphs. For a simple graph of order , we
show that and characterize the extremal graphs.
The Nordhaus-Gaddum-type results for the regular -independence number of
graphs are also obtained.Comment: 18 pages, 3 figures. arXiv admin note: text overlap with
arXiv:1306.5026 by other author
The Steiner diameter of a graph
The Steiner distance of a graph, introduced by Chartrand, Oellermann, Tian
and Zou in 1989, is a natural generalization of the concept of classical graph
distance. For a connected graph of order at least and , the \emph{Steiner distance} among the vertices of is the
minimum size among all connected subgraphs whose vertex sets contain . Let
be two integers with . Then the \emph{Steiner
-eccentricity } of a vertex of is defined by . Furthermore, the
\emph{Steiner -diameter} of is . In 2011, Chartrand, Okamoto and Zhang showed that . In this paper, graphs with are
characterized, respectively. We also consider the Nordhaus-Gaddum-type results
for the parameter . We determine sharp upper and lower bounds of
and
for a graph of order . Some graph classes attaining these bounds are
also given.Comment: 14 page
Further results on the global cyclicity index of graphs
Being motivated in terms of mathematical concepts from the theory of
electrical networks, Klein & Ivanciuc introduced and studied a new
graph-theoretic cyclicity index--the global cyclicity index (Graph cyclicity,
excess conductance, and resistance deficit, J. Math. Chem. 30 (2001) 271--287).
In this paper, by utilizing techniques from graph theory, electrical network
theory and real analysis, we obtain some further results on this new cyclicity
measure, including the strictly monotone increasing property, some lower and
upper bounds, and some Nordhuas-Gaddum-type results. In particular, we
establish a relationship between the global cyclicity index and the
cyclomatic number of a connected graph with vertices and
edges: Comment: 17pages, 1 figur
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