125 research outputs found
On Weakly Distinguishing Graph Polynomials
A univariate graph polynomial P(G;X) is weakly distinguishing if for almost
all finite graphs G there is a finite graph H with P(G;X)=P(H;X). We show that
the clique polynomial and the independence polynomial are weakly
distinguishing. Furthermore, we show that generating functions of induced
subgraphs with property C are weakly distinguishing provided that C is of
bounded degeneracy or tree-width. The same holds for the harmonious chromatic
polynomial
Some colouring problems for Paley graphs
The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved
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 sharp upper bound for the harmonious total chromatic number of graphs and multigraphs
A proper total colouring of a graph is called harmonious if it has the
further property that when replacing each unordered pair of incident vertices
and edges with their colours, then no pair of colours appears twice. The
smallest number of colours for it to exist is called the harmonious total
chromatic number of , denoted by . Here, we give a general upper
bound for in terms of the order of . Our two main results are
obvious consequences of the computation of the harmonious total chromatic
number of the complete graph and of the complete multigraph , where is the number of edges joining each pair of vertices of
. In particular, Araujo-Pardo et al. have recently shown that
. In this paper, we
prove that except for
and ; therefore, , for every graph on vertices. Finally, we
extend such a result to the harmonious total chromatic number of the complete
multigraph and as a consequence show that
for , where is a multigraph such that is the
maximum number of edges between any two vertices.Comment: 11 pages, 5 figure
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
- …