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) $L^p$ 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.

On H-irregularity Strengths of G-amalgamation of Graphs

A simple graph G=(V(G),E(G)) admits an H-covering if every edge in E(G) belongs at least to one subgraph of G isomorphic to a given graph H. Then the graph G admitting H-covering admits an H-irregular total k-labeling f: V(G) U E(G) \to {1, 2, ..., k} if for every two different subgraphs H\u27 and H\u27\u27 isomorphic to H there is $wt_{f}(H\u27) \neq wt_{f}(H\u27\u27)$, where $wt_{f}(H)= \sum \limits_{v\in V(H)} f(v) + \sum \limits_{e \in E(H)} f(e)$ is the associated H-weight. The minimum k for which the graph G has an H-irregular total k-labeling is called the total H-irregularity strength of the graph G.In this paper, we obtain the precise value of the total H-irregularity strength of G-amalgamation of graphs

Group twin coloring of graphs

For a given graph $G$, the least integer $k\geq 2$ such that for every Abelian group $\mathcal{G}$ of order $k$ there exists a proper edge labeling $f:E(G)\rightarrow \mathcal{G}$ so that $\sum_{x\in N(u)}f(xu)\neq \sum_{x\in N(v)}f(xv)$ for each edge $uv\in E(G)$ is called the \textit{group twin chromatic index} of $G$ and denoted by $\chi'_g(G)$. This graph invariant is related to a few well-known problems in the field of neighbor distinguishing graph colorings. We conjecture that $\chi'_g(G)\leq \Delta(G)+3$ for all graphs without isolated edges, where $\Delta(G)$ is the maximum degree of $G$, and provide an infinite family of connected graph (trees) for which the equality holds. We prove that this conjecture is valid for all trees, and then apply this result as the base case for proving a general upper bound for all graphs $G$ without isolated edges: $\chi'_g(G)\leq 2(\Delta(G)+{\rm col}(G))-5$, where ${\rm col}(G)$ denotes the coloring number of $G$. This improves the best known upper bound known previously only for the case of cyclic groups $\mathbb{Z}_k$