Browsing by Author "Hegde, S. M."

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A Study of Harmonious and Complete Colorings of Digraphs
(National Institute of Technology Karnataka, Surathkal, 2013) Hegde, S. M.; Shetty, Sudhakar
In this research work, we have extended the concept of harmonious colorings, complete colorings and set colorings of graphs to directed graphs. A harmonious coloring of any digraph D is an assignment of colors to the vertices of D and the color of an arc is defined to be the ordered pair of colors to its end vertices such that all arc colors are distinct. The proper harmonious coloring number is the least number of colors needed in such a coloring. Also, we obtain a lower bound for the proper harmonious coloring of any digraphs and regular digraphs and investigate the proper harmonious coloring number of some classes of digraphs. A complete coloring of a digraph D is a proper vertex coloring of D such that, for any ordered pair of colors, there is at least one arc of D whose endpoints are colored with this pair of colors. The achromatic number of D is the maximum number of colors in a proper complete coloring of D. We obtain an upper bound for the achromatic number of digraphs. Also, we find the achromatic number of some classes of digraphs. We have extended the concept of set colorings of graphs to set colorings of digraphs. We have given some necessary conditions for a digraph to admit a strong set coloring (proper set coloring). We will characterize strongly (properly) set colorable digraphs. Also, we find the construction of strongly (properly) set colorable directed caterpillars.
A Study on Graph Operators and Colorings
(National Institute of Technology Karnataka, Surathkal, 2017) Suresh Dara, V. V. P. R. V. B.; Hegde, S. M.
In 1972, Erdös - Faber - Lovász conjectured that, if H is a linear hypergraph consisting of n edges of cardinality n, then it is possible to color the vertices with n colors so that no two vertices with the same color are in the same edge. In this research work we give a method of coloring of the linear hypergraph H satisfying the hypothesis of the conjecture and we partially prove the Erdös - Faber - Lovász conjecture theoretically. 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 2 KG form an edge if and only if Qi \Qj 6= /0. 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. Let G be a labeled graph of order a, 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 2 F1 and f 2= 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, Fdivergence, F-depth and F-stability of any graph G.
A Study on Labelings of Directed Graphs
(National Institute of Technology Karnataka, Surathkal, 2017) Kumudakshi; Hegde, S. M.
In this thesis, two types of graph labeling problems has been studied namely, graceful and sequential labeling problems of digraphs. The use of modular arithmetic in these labeling ties them to a variety of algebraic problems. Using some of the algebraic structures such as (v; k; λ) di erence set, complete mapping and partition theory the gracefulness of some known class of digraphs has been proved. A construction of bigraphs from digraphs (vice-versa) are given using an adjacency matrix. Using this constructive method it is proved that, the graceful labelings of some class of bigraphs gives rise to graceful digraphs and vice-versa. Further, a relationship between the sequential labelings and graceful labelings of digraphs has been given. With this relation it is proved that, the sequential digraphs are related to near complete mappings and cyclic multiplicative groups. Also, for some more class of digraphs its gracefulness and sequentialness has been proved.