841 research outputs found
New bounds on the Grundy number of products of graphs
The Grundy number of a graph G is the largest k such that G has a greedy k-colouring, that is, a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we give new bounds on the Grundy number of the product of two graphs
New bounds on the Grundy number of products of graphs
International audienceThe Grundy number of a graph G is the largest k such that G has a greedy k- colouring, that is, a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we give new bounds on the Grundy number of the product of two graphs
Dominating sequences in grid-like and toroidal graphs
A longest sequence of distinct vertices of a graph such that each
vertex of dominates some vertex that is not dominated by its preceding
vertices, is called a Grundy dominating sequence; the length of is the
Grundy domination number of . In this paper we study the Grundy domination
number in the four standard graph products: the Cartesian, the lexicographic,
the direct, and the strong product. For each of the products we present a lower
bound for the Grundy domination number which turns out to be exact for the
lexicographic product and is conjectured to be exact for the strong product. In
most of the cases exact Grundy domination numbers are determined for products
of paths and/or cycles.Comment: 17 pages 3 figure
First-Fit coloring of Cartesian product graphs and its defining sets
Let the vertices of a Cartesian product graph be ordered by an
ordering . By the First-Fit coloring of we mean the
vertex coloring procedure which scans the vertices according to the ordering
and for each vertex assigns the smallest available color. Let
be the number of colors used in this coloring. By
introducing the concept of descent we obtain a sufficient condition to
determine whether , where and
are arbitrary orders. We study and obtain some bounds for , where is any quasi-lexicographic ordering. The First-Fit
coloring of does not always yield an optimum coloring. A
greedy defining set of is a subset of vertices in the
graph together with a suitable pre-coloring of such that by fixing the
colors of the First-Fit coloring of yields an optimum
coloring. We show that the First-Fit coloring and greedy defining sets of
with respect to any quasi-lexicographic ordering (including the known
lexicographic order) are all the same. We obtain upper and lower bounds for the
smallest cardinality of a greedy defining set in , including some
extremal results for Latin squares.Comment: Accepted for publication in Contributions to Discrete Mathematic
Grundy dominating sequences and zero forcing sets
In a graph a sequence of vertices is Grundy
dominating if for all we have and is Grundy total dominating if for all
we have .
The length of the longest Grundy (total) dominating sequence has
been studied by several authors. In this paper we introduce two
similar concepts when the requirement on the neighborhoods is
changed to or
. In the former case we
establish a strong connection to the zero forcing number of a graph,
while we determine the complexity of the decision problem in the
latter case. We also study the relationships among the four
concepts, and discuss their computational complexities
On The Equality Of The Grundy Numbers Of A Graph
Our work becomes integrated into the general problem of the stability of the
network ad hoc. Some, works attacked this problem. Among these works, we find
the modelling of the network ad hoc in the form of a graph. Thus the problem of
stability of the network ad hoc which corresponds to a problem of allocation of
frequency amounts to a problem of allocation of colors in the vertex of graph.
we present use a parameter of coloring the number of Grundy. The Grundy number
of a graph G, denoted by (G), is the largest k such that G has a greedy
k-coloring, that is a coloring with colours obtained by applying the greedy
algorithm according to some ordering of the vertices of G. In this paper, we
study the Grundy number of the lexicographic, Cartesian and direct products of
two graphs in terms of the Grundy numbers of these graphs.Comment: 13 pages, International Journal of Next-Generation Networks
(IJNGN),December 201
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