877 research outputs found
A Unified View of Graph Regularity via Matrix Decompositions
We prove algorithmic weak and \Szemeredi{} regularity lemmas for several
classes of sparse graphs in the literature, for which only weak regularity
lemmas were previously known. These include core-dense graphs, low threshold
rank graphs, and (a version of) upper regular graphs. More precisely, we
define \emph{cut pseudorandom graphs}, we prove our regularity lemmas for these
graphs, and then we show that cut pseudorandomness captures all of the above
graph classes as special cases.
The core of our approach is an abstracted matrix decomposition, roughly
following Frieze and Kannan [Combinatorica '99] and \Lovasz{} and Szegedy
[Geom.\ Func.\ Anal.\ '07], which can be computed by a simple algorithm by
Charikar [AAC0 '00]. This gives rise to the class of cut pseudorandom graphs,
and using work of Oveis Gharan and Trevisan [TOC '15], it also implies new
PTASes for MAX-CUT, MAX-BISECTION, MIN-BISECTION for a significantly expanded
class of input graphs. (It is NP Hard to get PTASes for these graphs in
general.
Total Edge Irregularity Strength for Graphs
An edge irregular total -labelling of a graph is a labelling of the vertices and the edges of
in such a way that any two different edges have distinct weights. The
weight of an edge , denoted by , is defined as the sum of the label
of and the labels of two vertices which incident with , i.e. if ,
then . The minimum for which has an edge
irregular total -labelling is called the total edge irregularity strength of
In this paper, we determine total edge irregularity of connected and
disconnected graphs
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