4,046 research outputs found
Harmonious Coloring of Trees with Large Maximum Degree
A harmonious coloring of is a proper vertex coloring of such that
every pair of colors appears on at most one pair of adjacent vertices. The
harmonious chromatic number of , , is the minimum number of colors
needed for a harmonious coloring of . We show that if is a forest of
order with maximum degree , then h(T)=
\Delta(T)+2, & if $T$ has non-adjacent vertices of degree $\Delta(T)$;
\Delta(T)+1, & otherwise.
Moreover, the proof yields a polynomial-time algorithm for an optimal
harmonious coloring of such a forest.Comment: 8 pages, 1 figur
Upward Three-Dimensional Grid Drawings of Graphs
A \emph{three-dimensional grid drawing} of a graph is a placement of the
vertices at distinct points with integer coordinates, such that the straight
line segments representing the edges do not cross. Our aim is to produce
three-dimensional grid drawings with small bounding box volume. We prove that
every -vertex graph with bounded degeneracy has a three-dimensional grid
drawing with volume. This is the broadest class of graphs admiting
such drawings. A three-dimensional grid drawing of a directed graph is
\emph{upward} if every arc points up in the z-direction. We prove that every
directed acyclic graph has an upward three-dimensional grid drawing with
volume, which is tight for the complete dag. The previous best upper
bound was . Our main result is that every -colourable directed
acyclic graph ( constant) has an upward three-dimensional grid drawing with
volume. This result matches the bound in the undirected case, and
improves the best known bound from for many classes of directed
acyclic graphs, including planar, series parallel, and outerplanar
A note on "Folding wheels and fans."
In S.Gervacio, R.Guerrero and H.Rara, Folding wheels and fans, Graphs and
Combinatorics 18 (2002) 731-737, the authors obtain formulas for the clique
numbers onto which wheels and fans fold. We present an interpolation theorem
which generalizes their theorems 4.2 and 5.2. We show that their formula for
wheels is wrong. We show that for threshold graphs, the achromatic number and
folding number coincides with the chromatic number
Asymmetric coloring games on incomparability graphs
Consider the following game on a graph : Alice and Bob take turns coloring
the vertices of properly from a fixed set of colors; Alice wins when the
entire graph has been colored, while Bob wins when some uncolored vertices have
been left. The game chromatic number of is the minimum number of colors
that allows Alice to win the game. The game Grundy number of is defined
similarly except that the players color the vertices according to the first-fit
rule and they only decide on the order in which it is applied. The -game
chromatic and Grundy numbers are defined likewise except that Alice colors
vertices and Bob colors vertices in each round. We study the behavior of
these parameters for incomparability graphs of posets with bounded width. We
conjecture a complete characterization of the pairs for which the
-game chromatic and Grundy numbers are bounded in terms of the width of
the poset; we prove that it gives a necessary condition and provide some
evidence for its sufficiency. We also show that the game chromatic number is
not bounded in terms of the Grundy number, which answers a question of Havet
and Zhu
On the oriented chromatic number of dense graphs
Let be a graph with vertices, edges, average degree , and maximum degree . The \emph{oriented chromatic number} of is the maximum, taken over all orientations of , of the minimum number of colours in a proper vertex colouring such that between every pair of colour classes all edges have the same orientation. We investigate the oriented chromatic number of graphs, such as the hypercube, for which . We prove that every such graph has oriented chromatic number at least . In the case that , this lower bound is improved to . Through a simple connection with harmonious colourings, we prove a general upper bound of \Oh{\Delta\sqrt{n}} on the oriented chromatic number. Moreover this bound is best possible for certain graphs. These lower and upper bounds are particularly close when is ()-regular for some constant , in which case the oriented chromatic number is between and
Topological and algebraic properties of universal groups for right-angled buildings
We study universal groups for right-angled buildings. Inspired by Simon
Smith's work on universal groups for trees, we explicitly allow local groups
that are not necessarily finite nor transitive. We discuss various topological
and algebraic properties in this extended setting. In particular, we
characterise when these groups are locally compact, when they are abstractly
simple, when they act primitively on residues of the building, and we discuss
some necessary and sufficient conditions for the groups to be compactly
generated.
We point out that there are unexpected aspects related to the geometry and
the diagram of these buildings that influence the topological and algebraic
properties of the corresponding universal groups.Comment: 31 page
Group Sum Chromatic Number of Graphs
We investigate the \textit{group sum chromatic number} (\gchi(G)) of
graphs, i.e. the smallest value such that taking any Abelian group \gr of
order , there exists a function f:E(G)\rightarrow \gr such that the sums
of edge labels properly colour the vertices. It is known that
\gchi(G)\in\{\chi(G),\chi(G)+1\} for any graph with no component of order
less than and we characterize the graphs for which \gchi(G)=\chi(G).Comment: Accepted for publication in European Journal of Combinatorics,
Elsevie
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