Let G be a simple graph on n vertices with list chromatic number χl = s. If each vertex of G is assigned a list of t colours Albertson, Grossman and Haas [1] asked how many of the vertices, λ t,s, are necessarily colourable from these lists? They conjectured that λt,s ≥ tn/s. Their work was extended by Chappell [2]. We improve the known lower bounds for λt,s. Primary AMS subject classification:...

Characterization of Pre-Monsoon Aerosol and Aerosol-Cloud-Rainfall Interactions in Central Nepal by Prabhakar Shrestha Department of Civil and Environmental Engineering Duke University Date:_______________________ Approved: ___________________________ Ana P. Barros, Supervisor ___________________________ Andrei Khlystov ___________________________ John D. Albertson ___________________________ M...

Pamela L. Paris, Armann Andaya, Jane Fridlyand, Ajay N. Jain, Vivian Weinberg, David Kowbel, John H. Brebner, Jeff Simko, J.E. Vivienne Watson, Stas Volik, Donna G. Albertson, Daniel Pinkel, Janneke C. Alers, Theodorus H. van der Kwast, Kees J. Vissers, Fritz H. Schroder, Mark F. Wildhagen, Phillip G. Febbo, Arul M. Chinnaiyan, Kenneth J. Pienta, Peter R. Carroll, Mark A. Rubin, Colin Collins* ...

Albertson and Collins defined the distinguishing number of a graph to be the smallest number of colors needed to color its vertices so that the coloring is preserved only by the identity automorphism. Collins and Trenk followed by defining the distinguishing chromatic number of a graph to be the smallest size of a coloring that is both proper and distinguishing. We show that, with two exception...

Given a directed graph, an acyclic set is a set of vertices inducing a directed subgraph with no directed cycle. In this note we show that for all integers n ≥ g ≥ 3, there exist oriented planar graphs of order n and digirth g for which the size of the maximum acyclic set is at most dn(g−2)+1 g−1 e. When g = 3 this result disproves a conjecture of Harutyunyan and shows that a question of Albert...

Land-atmosphere Interaction: from Atmospheric Boundary Layer to Soil Moisture Dynamics by Jun Yin Department of Civil and Environmental Engineering Duke University Date:_______________________ Approved: ___________________________ John Albertson, Supervisor ___________________________ Amilcare Porporato ___________________________ Gabriel Katul ___________________________ J.R. Rigby An abstract...

We show that if G is a graph such that every edge is in at least two triangles, then G contains a spanning tree with no vertex of degree 2 (a homeomorphically irreducible spanning tree). This result was originally asked in a question format by Albertson, Berman, Hutchinson, and Thomassen in 1979, and then conjectured to be true by Archdeacon in 2009. MSC2010 : 05C05, 05C75

The irregularity of a simple undirected graph G was defined by Albertson [5] as irr(G) = ∑ uv∈E(G) |dG(u)− dG(v)|, where dG(u) denotes the degree of a vertex u ∈ V (G). In this paper we consider the irregularity of graphs under several graph operations including join, Cartesian product, direct product, strong product, corona product, lexicographic product, disjunction and symmetric difference. ...