N. Narayanan

Bio: N. Narayanan is an academic researcher from Indian Institute of Technology Madras. The author has contributed to research in topics: Edge coloring & Bipartite graph. The author has an hindex of 10, co-authored 35 publications receiving 450 citations. Previous affiliations of N. Narayanan include Indian Institutes of Technology & National Taiwan University.

Regularity of powers of bipartite graphs

Abstract: Let G be a finite simple graph and I(G) denote the corresponding edge ideal. For all $$s \ge 1$$ , we obtain upper bounds for $${\text {reg}}(I(G)^s)$$ for bipartite graphs. We then compare the properties of G and $$G'$$ , where $$G'$$ is the graph associated with the polarization of the ideal $$(I(G)^{s+1} : e_1\cdots e_s)$$ , where $$e_1,\cdots , e_s$$ are edges of G. Using these results, we explicitly compute $${\text {reg}}(I(G)^s)$$ for several subclasses of bipartite graphs.

Acyclic Edge Colouring of Outerplanar Graphs

Abstract: An acyclicedge colouring of a graph is a proper edge colouring having no 2-coloured cycle, that is, a colouring in which the union of any two colour classes forms a linear forest. The acyclic chromatic indexof a graph is the minimum number ksuch that there is an acyclic edge colouring using kcolours and is usually denoted by ai¾?(G). Determining ai¾?(G) exactly is a very hard problem (both theoretically and algorithmically) and is not determined even for complete graphs. We show that ai¾?(G) ≤ Δ(G) + 1, if Gis an outerplanar graph. This bound is tight within an additive factor of 1 from optimality. Our proof is constructive leading to an $O\!\left({n \log \Delta}\right)$ time algorithm. Here, Δ= Δ(G) denotes the maximum degree of the input graph.

Regularity of Powers of Bipartite Graphs

Abstract: Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph associated with the polarization of the ideal $(I(G)^{s+1} : e_1\cdots e_s)$, where $e_1,\ldots e_s$ are edges of $G$. Using these results, we explicitly compute reg$(I(G)^s)$ for several subclasses of bipartite graphs.

Strong Chromatic Index of 2‐Degenerate Graphs

Abstract: We prove that the strong chromatic index of a 2-degenerate graph is linear in the maximum degree Δ. This includes the class of all chordless graphs (graphs in which every cycle is induced) which in turn includes graphs where the cycle lengths are multiples of four, and settles a problem by Faudree et al. (Ars Combin 29(B) (1990), 205–211). © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 119–126, 2013

Phd by thesis

Abstract: Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Random graphs

Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

A Theory of Graphs

Abstract: A graph is just a bunch of points with lines between some of them, like a map of cities linked by roads. A rather simple notion. Nevertheless, the theory of graphs has broad and important applications, because so many things can be modeled by graphs. For example, planar graphs — graphs in which none of the lines cross are— important in designing computer chips and other electronic circuits. Also, various puzzles and games are solved easily if a little graph theory is applied.

New Constructive Aspects of the Lovasz Local Lemma

Abstract: The Lov\'{a}sz Local Lemma (LLL) states that the probability that none of a set of "bad" events happens is nonzero if the probability of each event is small compared to the number of bad events it depends on. A series of results have provided algorithms to efficiently construct structures whose existence is (non-constructively) guaranteed by the full asymmetric LLL, culminating in the recent breakthrough of Moser & Tardos. We show that the output distribution of the Moser-Tardos procedure has sufficient randomness, leading to two classes of algorithmic applications. We first show that when an LLL application provides a small amount of slack, the running time of the Moser-Tardos algorithm is polynomial in the number of underlying independent variables (not events!), and can thus be used to give efficient constructions in cases where the underlying proof applies the LLL to super-polynomially many events (or where finding a bad event that holds is computationally hard). We demonstrate our method on applications including: the first constant-factor approximation algorithm for the Santa Claus problem, as well as efficient algorithms for acyclic edge coloring, non-repetitive graph colorings, and Ramsey-type graphs. Second, we show applications to cases where a few of the bad events can hold, leading to the first such algorithmic applications of the LLL: MAX $k$-SAT is an illustrative example of this.