148 research outputs found
Labeling of graphs, sumset of squares of units modulo n and resonance varieties of matroids
This thesis investigates problems in a number of different areas of graph theory and its applications in other areas of mathematics. Motivated by the 1-2-3-Conjecture, we consider the closed distinguishing number of a graph G, denoted by dis[G]. We provide new upper bounds for dis[G] by using the Combinatorial Nullstellensatz. We prove that it is NP-complete to decide for a given planar subcubic graph G, whether dis[G] = 2. We show that for each integer t there is a bipartite graph G such that dis[G] \u3e t. Then some polynomial time algorithms and NP-hardness results for the problem of partitioning the edges of a graph into regular and/or locally irregular subgraphs are presented. We then move on to consider Johnson graphs to find resonance varieties of some classes of sparse paving matroids. The last application we consider is in number theory, where we find the number of solutions of the equation x21 + _ _ _ + x2 k = c, where c 2 Zn, and xi are all units in the ring Zn. Our approach is combinatorial using spectral graph theory
Graph Coloring via Degeneracy in Streaming and Other Space-Conscious Models
We study the problem of coloring a given graph using a small number of colors
in several well-established models of computation for big data. These include
the data streaming model, the general graph query model, the massively parallel
computation (MPC) model, and the CONGESTED-CLIQUE and the LOCAL models of
distributed computation. On the one hand, we give algorithms with sublinear
complexity, for the appropriate notion of complexity in each of these models.
Our algorithms color a graph using about colors, where
is the degeneracy of : this parameter is closely related to the
arboricity . As a function of alone, our results are
close to best possible, since the optimal number of colors is .
On the other hand, we establish certain lower bounds indicating that
sublinear algorithms probably cannot go much further. In particular, we prove
that any randomized coloring algorithm that uses many colors,
would require storage in the one pass streaming model, and
many queries in the general graph query model, where is the
number of vertices in the graph. These lower bounds hold even when the value of
is known in advance; at the same time, our upper bounds do not
require to be given in advance.Comment: 26 page
- …