Browsing by Author "Hegde, S.M."
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Item A Partial Solution to Cordial Tree Conjecture(Taru Publications, 2014) Hegde, S.M.; Murthy, T.S.Abstract: In this paper, we prove the weak harmonious tree conjecture by Andrzej ?ak (2009) using the value sets of polynomials. Consequently, it partially proves the cordial tree conjecture by Mark Hovey (1991), that is all trees of order n < p are p-cordial, where p is a prime. © 2014, © Taru Publications.Item A Partial Solution to Linear Congruence Conjecture(Springer India sanjiv.goswami@springer.co.in, 2016) Hegde, S.M.; Murthy, T.S.Adams and Ponomarenko (Involv J Math 3(3):341–344, 2010) conjectured that when n is composite, ki? Zn satisfying (Formula Presented). In this paper, distinct solution has been constructed to the linear congruence when ?i=1kai=n-1, using super edge-magic labeling of trees. © 2016, The National Academy of Sciences, India.Item Achromatic number of some classes of digraphs(World Scientific, 2024) Hegde, S.M.; Castelino, L.P.Let D be a directed graph with n vertices and m arcs. A function g: V (D) →{1, 2, ⋯, k} where k ≤ n is called a complete coloring of D if and only if for every arc uv of D, the ordered pair (g(u), g(v)) appears at least once. If the pair (i, i) is not assigned, then g is called a proper complete coloring of D. The maximum k for which D admits a proper complete coloring is called the achromatic number of D and is denoted by ψc →(D). We obtain the upper bound for the achromatic number of digraphs and regular digraphs and investigate the same for some classes of digraphs such as unipath, unicycle, circulant digraphs, etc. © 2024 World Scientific Publishing Company.Item Acyclic chromatic index of chordless graphs(Elsevier B.V., 2023) Basavaraju, M.; Hegde, S.M.; Kulamarva, S.An acyclic edge coloring of a graph is a proper edge coloring with no bichromatic cycles. The acyclic chromatic index of a graph G denoted by a′(G), is the minimum integer k such that G has an acyclic edge coloring with k colors. It was conjectured by Fiamčík [13] that a′(G)≤Δ+2 for any graph G with maximum degree Δ. Linear arboricity of a graph G, denoted by la(G), is the minimum number of linear forests into which the edges of G can be partitioned. A graph is said to be chordless if no cycle in the graph contains a chord. By a result of Basavaraju and Chandran [6], if G is chordless, then a′(G)≤Δ+1. Machado, de Figueiredo and Trotignon [23] proved that the chromatic index of a chordless graph is Δ when Δ≥3. We prove that for any chordless graph G, a′(G)=Δ, when Δ≥3. Notice that this is an improvement over the result of Machado et al., since any acyclic edge coloring is also a proper edge coloring and we are using the same number of colors. As a byproduct, we prove that [Formula presented], when Δ≥3. To obtain the result on acyclic chromatic index, we prove a structural result on chordless graphs which is a refinement of the structure given by Machado et al. [23] in case of chromatic index. This might be of independent interest. © 2023 Elsevier B.V.Item An improved upper bound for the domination number of a graph(Springer, 2025) Arumugam, S.; Hegde, S.M.; Kulamarva, S.Let G be a graph of order n. A classical upper bound for the domination number of a graph G having no isolated vertices is ?n2?. However, for several families of graphs, we have ?(G)??n? which gives a substantially improved upper bound. In this paper, we give a condition necessary for a graph G to have ?(G)??n?, and some conditions sufficient for a graph G to have ?(G)??n?. We also present a characterization of all connected graphs G of order n with ?(G)=?n?. Further, we prove that for a graph G not satisfying rad(G)=diam(G)=rad(G¯)=diam(G¯)=2, deciding whether ?(G)??n? or ?(G¯)??n? can be done in polynomial time. We conjecture that this decision problem can be solved in polynomial time for any graph G. © Indian Academy of Sciences 2025.Item 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.Item CFD analysis of turboprop engine oil cooler duct for best rate of climb condition(2016) Kalia, S.; Ca, V.; Hegde, S.M.Turboprop engines are widely used in commuter category airplanes. Aircraft Design bureaus routinely conduct the flight tests to confirm the performance of the system. The lubrication system of the engine is designed to provide a constant supply of clean lubrication oil to the engine bearings, the reduction gears, the torque-meter, the propeller and the accessory gearbox. The oil lubricates, cools and also conducts foreign material to the oil filter where it is removed from further circulation. Thus a means of cooling the engine oil must be provided and a suitable oil cooler (OC) and ducting system was selected and designed for this purpose. In this context, it is relevant to study and analyse behaviour of the engine oil cooler system before commencing actual flight tests. In this paper, the performance of the oil cooler duct with twin flush NACA inlet housed inside the nacelle has been studied for aircraft best rate of climb (ROC) condition using RANS based SST K-omega model by commercial software ANSYS Fluent 13.0. From the CFD analysis results, it is found that the mass flow rate captured and pressure drop across the oil cooler for the best ROC condition is meeting the oil cooler manufacturer requirements thus, the engine oil temperature is maintained within prescribed limits. � Published under licence by IOP Publishing Ltd.Item CFD analysis of turboprop engine oil cooler duct for best rate of climb condition(Institute of Physics Publishing michael.roberts@iop.org, 2016) Kalia, S.; Ca, V.; Hegde, S.M.Turboprop engines are widely used in commuter category airplanes. Aircraft Design bureaus routinely conduct the flight tests to confirm the performance of the system. The lubrication system of the engine is designed to provide a constant supply of clean lubrication oil to the engine bearings, the reduction gears, the torque-meter, the propeller and the accessory gearbox. The oil lubricates, cools and also conducts foreign material to the oil filter where it is removed from further circulation. Thus a means of cooling the engine oil must be provided and a suitable oil cooler (OC) and ducting system was selected and designed for this purpose. In this context, it is relevant to study and analyse behaviour of the engine oil cooler system before commencing actual flight tests. In this paper, the performance of the oil cooler duct with twin flush NACA inlet housed inside the nacelle has been studied for aircraft best rate of climb (ROC) condition using RANS based SST K-omega model by commercial software ANSYS Fluent 13.0. From the CFD analysis results, it is found that the mass flow rate captured and pressure drop across the oil cooler for the best ROC condition is meeting the oil cooler manufacturer requirements thus, the engine oil temperature is maintained within prescribed limits. © Published under licence by IOP Publishing Ltd.Item Combinatorial labelings of graphs(2006) Hegde, S.M.; Shetty, S.A (p, g)-graph G is said to be a permutation (combination) graph if G admits an assignment of distinct integers 1, 2, 3, ..., p to the vertices such that edge values obtained by the number of permutations (combinations) of larger vertex value taken smaller vertex value at a time are distinct. In this paper we obtain a necessary condition for combination graph and study structure of permutation and combination graphs which includes some open problems.Item CONSTRUCTION AND ANALYSIS OF GRAPH MODELS FOR MULTIPROCESSOR INTERCONNECTION NETWORKS(Faculty of Organizational Sciences, Belgrade, 2022) Hegde, S.M.; Saumya, Y.M.A graph G can serve as a model for the Multiprocessor Interconnection Net- works (MINs) in which the vertices represent the processors, while the edges represent connections between processors. This paper presents several graphs that could qualify as models for efficient MINs based on the small values of the graph tightness previously introduced by Cvetkovic and Davidovic in 2008. These graphs are constructed using some well-known and widely used graph operations. The tightness values of these graphs range from O( 4 √ N) to O( √ N), where N is the order of the graph under consideration. Also, two new graph tightness values, namely Third type mixed tightness t3(G) and Second type of Structural tightness t4(G) are defined in this paper. It has been shown that these tightness types are easier to calculate than the others for the considered graphs. Moreover, their values are significantly smaller. © 2022 Faculty of Organizational Sciences, Belgrade. All rights reserved.Item Construction of graceful digraphs using algebraic structures(Taru Publications, 2016) Hegde, S.M.; Kumudakshi, K.Abstract: In the early 1980?s Bloom and Hsu extended the notation of graceful labelings to directed graphs, and gave a relationship between graceful digraphs and a variety of algebraic structures. In this paper using a cyclic (v, k, ?) difference set with ? copies of elements of Zv\ {0}, we construct graceful digraphs of k vertices and v – 1 arcs. It is known that if gracefully labelled graph has e edges then its symmetric digraph is graceful with the same vertex labels. Although, the cycle Cm is not graceful for m?1, 2 (mod 4) we show that the symmetric digraph based on cycle Cm i.e the double cycle, DCm which is constructed from a m-cycle by replacing each edge by a pair of arcs, edge xy gives rise to arcs (x, y) and (y, x), is graceful for any m vertices specifically for m?1, 2 (mod 4). © 2016 TARU Publications.Item 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.Item Further Results on Graceful Digraphs(Springer, 2016) Hegde, S.M.; ShivarajkumarA digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from { 0 , 1 ,.. , q} to each vertex v. The vertex values, in turn, induce a value g(u, v) on each arc (u, v) where g(u,v)=(g(v)-g(u))(modq+1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. In this paper, we prove a general result on graceful digraphs of which Du and Sun’s conjecture (J. Beijing Univ. Posts Telecommun, 17: 85–88 1994) is a special case. Further, we provide an upper bound for the number of non isomorphic graceful directed cycles obtained from a graceful labeling of the unicycle C n ?. © 2015, Springer India Pvt. Ltd.Item Further Results on Graceful Directed Graphs(Elsevier B.V., 2016) Hegde, S.M.; Kumudakshi, K.In this paper we present the gracefulness of the directed graph Pm?Pn? which is an orientation of the planar grid graph Pm?Pn, in which each cell is a unicycle of length four. © 2016 Elsevier B.V.Item Further Results on Harmonious Colorings of Digraphs(2011) Hegde, S.M.; Castelino, L.P.Let D be a directed graph with n vertices and m edges. A function f: V (D) ? {1, 2, 3, ..., t}, where t ? n is said to be a harmonious coloring of D if for any two edges xy and uv of D, the ordered pair (f(x), f(y)) ? (f(u), f(v)). If no pair (i, i) is assigned, then f is said to be a proper harmonious coloring of D. The minimum t for which D admits a proper harmonious coloring is called the proper harmonious coloring number of D. We investigate the proper harmonious coloring number of graphs such as alternating paths and alternating cycles.Item Further results on proper and strong set colorings of graphs(2012) Hegde, S.M.; Sumana, M.K.A set coloring ? of a graph G is defined as an assignment of distinct subsets of a finite set X of colors to the vertices of G such that all the colors of the edges which are obtained as the symmetric differences of the sets assigned to their end-vertices are distinct. Additionally, if all the sets on the vertices and edges of G form the set of all nonempty subsets of X, then the coloring ? is said to be a strong set coloring, and the graph G is called strongly set colorable. If all the nonempty subsets of X are obtained on the edges of G, then ? is called a proper set coloring, and such a graph G is called properly set colorable. The set coloring number of a graph G, denoted by ?(G), is the smallest cardinality of a set X such that G has a set coloring with respect to X. This paper discusses the set coloring number of certain classes of graphs and the construction of strongly set colorable caterpillars which are also properly set colorable. An upper bound for b is found for K 3,b to admit set coloring with set coloring number n.