You can download some of my papers(in pdf form) here.
They can be viewed and printed with the free Adobe Acrobat Reader.
Click here
to get the latest version.)

**Bounds for Partial List
Colourings****
(in Ars Comb.),
R. Haas, D. Hanson and G. MacGillivray
Let G be a simple graph on n vertices with list chromatic number
s. If each vertex of G is assigned a list of t colours
Albertson, Grossman and Haas [1] asked how many of the vertices, l_{t,s},
are
necessarily colourable from these lists? They conjectured that l_{t,s}
>=
tn/s. Their work was extended by Chappell [2]. We improve the known
lower bounds for l_{t,s}.
**

**
Bases for Splines on a Subdivided Domain
(in Acta Math.)
Ruth Haas
Let S^{r}(D) be the module of all splines of smoothness
r on the rectilinear partition D which subdivides some
domain D. Further, let S^{r}(G) be the module of
all splines of smoothness r on G which also subdivides
D, where G is a finer subdivision of D.
We study the relationship between a generating set of S^{r}(D)
and a generating set for S^{r}(G) . This paper
gives an algorithm for extending a generating set for S^{r}(D)
to one for for S^{r}(G) . This method is built
on algebraic properties of splines and the Grobner Basis Algorithm.**

**Characterizations of Arboricity of
Graphs**(in Ars Comb.)
Ruth Haas

The aim of this paper is to give several characterizations for the following two
classes of graphs: (i) graphs for which adding *any* l edges produces a graph
which is decomposible into k spanning trees and (ii) graphs for which adding *some*
l edges produces a graph which is decomposible into k spanning trees.

**
Partial List Colorings**( in Discrete
Mathematics)
Michael O. Albertson, Sara Grossman and Ruth Haas

Suppose G is an s-choosable graph with n vertices, and every vertex of G is assigned a list of t colors. We conjecture that at least (t/s) n of the vertices of G can be colored from these lists.

We provide lower bounds and consider related questions. For instance we show that if G is X-colorable (rather than being s-choosable), then more than [1 - ({X - 1}/X)^t]n of the vertices of G can be colored from the lists and that this is asymptotically best possible. We include a number of open questions.

Abstract: A *star forest* of a graph G is a spanning subgraph of G in which
each component is a star. The minimum number of edges required to guarantee that
an arbitrary graph, or a bipartite graph, has a star forest of size n is determined.
Sharp lower bounds on the size of a largest star forest are also determined. For
bipartite graphs, these are used to obtain an upper bound on the domination number
in terms of the number of vertices and edges in the graph, which is an improvement
on a bound of Vizing. In turn, the results on bipartite graphs are used to determine
the minimum number of lattice points required so that there exists a subset of n
lattice points, no three of which form a right triangle with legs parallel to the
coordinate axes.

**Bounds on the
signed domination number
of a graph**.
Ruth Haas and Thomas B. Wexler

Abstract: Let G=(V,E) be a simple graph on vertex set V and define a
function f: V->{-1,1}. The function f is a signed dominating function
if for every vertex xe V, the closed
neighborhood of x contains more vertices with function value 1 than with
-1. The signed domination number of G, g_{s}(G), is the minimum weight of a signed
dominating function on G. We give a sharp lower bound on the signed
domination number of a general graph with a given minimum and maximum
degree, generalizing a number of previously known results. Using similar
techniques we give upper and lower bounds for the signed domination number
of some simple graph products: the grid P_{j}xP_{k
}C_{j}xP_{k} and C_{j}xC_{k}. For
fixed width, these bounds differ by only a constant.

**
Signed domination number
of a graph and its complement**.
Ruth Haas and Thomas B. Wexler

Abstract: Let G=(V,E) be a simple graph on vertex set V and define a
function f: V->{-1,1}. The function f is a signed dominating function
if for every vertex xe V, the closed
neighborhood of x contains more vertices with function value 1 than with
-1. The signed domination number of G, g_{s}(G), is the minimum weight of a signed
dominating function on G. Let G* denote the complement of G. In this paper
we establish upper and lower bounds on g_{s}(G) + g_{s}(G*) and |g_{s}(G)| |g_{s}(G*)|.