Faculty Publications

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Author: search.filters.author.Dara, S.Has files: No

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    The dynamics of the forest graph operator
    (University of Zielona Gora, 2016) Dara, S.; Hegde, S.M.; Deva, V.; Rao, S.B.; Zaslavsky, T.
    In 1966, Cummins introduced the "tree graph": the tree graph T(G) of a graph G (possibly infinite) has all its spanning trees as vertices, and distinct such trees correspond to adjacent vertices if they differ in just one edge, i.e., two spanning trees T1 and T2 are adjacent if T2 = T1 - e + f for some edges e ? T1 and f ? T1. The tree graph of a connected graph need not be connected. To obviate this difficulty we define the "forest graph": let G be a labeled graph of order ?, finite or infinite, and let N(G) be the set of all labeled maximal forests of G. The forest graph of G, denoted by F(G), is the graph with vertex set N(G) in which two maximal forests F1, F2 of G form an edge if and only if they differ exactly by one edge, i.e., F2 = F1 - e + f for some edges e ? F1 and f ? F1. Using the theory of cardinal numbers, Zorn's lemma, transfinite induction, the axiom of choice and the well-ordering principle, we determine the F-convergence, F-divergence, F-depth and F-stability of any graph G. In particular it is shown that a graph G (finite or infinite) is F-convergent if and only if G has at most one cycle of length 3. The F-stable graphs are precisely K3 and K1. The F-depth of any graph G different from K3 and K1 is finite. We also determine various parameters of F(G) for an infinite graph G, including the number, order, size, and degree of its components. © 2016, University of Zielona Gora. All rights reserved.
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    On clique convergence of graphs
    (Kalasalingam University info@kalasalingam.ac.in, 2016) Hegde, S.M.; Dara, S.
    Let G be a graph and KG be the set of all cliques of G, then the clique graph of G denoted by K(G) is the graph with vertex set KG and two elements Qi,Qj?KG form an edge if and only if Qi?Qj?0?. Iterated clique graphs are defined by K0(G)=G, and Kn(G)=K(Kn?1(G)) for n>0. In this paper we prove a necessary and sufficient condition for a clique graph K(G) to be complete when G=G1+G2, give a partial characterization for clique divergence of the join of graphs and prove that if G1, G2 are Clique-Helly graphs different from K1 and G=G1?G2, then K2(G)=G. © 2016 Kalasalingam University
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    Further results on Erd?s–Faber–Lovász conjecture
    (Taylor and Francis Ltd., 2020) Hegde, S.M.; Dara, S.
    In 1972, Erd?s–Faber–Lovász (EFL) conjectured that, if (Formula presented.) is a linear hypergraph consisting of (Formula presented.) edges of cardinality (Formula presented.), then it is possible to color the vertices with (Formula presented.) colors so that no two vertices with the same color are in the same edge. In 1978, Deza, Erdös and Frankl had given an equivalent version of the same for graphs: Let (Formula presented.) denote a graph with (Formula presented.) complete graphs (Formula presented.) (Formula presented.), each having exactly (Formula presented.) vertices and have the property that every pair of complete graphs has at most one common vertex, then the chromatic number of (Formula presented.) is (Formula presented.). The clique degree (Formula presented.) of a vertex (Formula presented.) in (Formula presented.) is given by (Formula presented.). In this paper we give a method for assigning colors to the graphs satisfying the hypothesis of the Erd?s–Faber–Lovász conjecture and every (Formula presented.) ((Formula presented.)) has atmost (Formula presented.) vertices of clique degree greater than one using Symmetric latin Squares and clique degrees of the vertices of (Formula presented.). © 2018 Kalasalingam University. Published with license by Taylor & Francis Group, LLC.
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    Bounds on Erd?s - Faber - Lovász conjecture - the uniform and regular cases
    (Indonesian Combinatorics Society, 2025) Hegde, S.M.; Dara, S.
    We consider the Erd?s - Faber - Lovász (EFL) conjecture for hypergraphs. This paper gives an upper bound for the chromatic number of r regular linear hypergraphs H of size n. If r ? 4, ?(H) ? 1.181n and if r = 3, ?(H) ? 1.281n. © (2025), (Indonesian Combinatorics Society). All rights reserved.