Faculty Publications
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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 Harmonious colorings of digraphs(Charles Babbage Research Centre, 2015) 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, .?} where ? ? n is said to be harmonious coloring of D if for any two edges xy and u? of D, the ordered pair (f(x), f(y)) ? (f(u), f(?)). If the pair (i, i) is not assigned, then / is said to be a proper harmonious coloring of D. The minimum ? is called the proper harmonious coloring number of D. We investigate the proper harmonious coloring number of graphs such as unidirectional paths, unicycles, inspoken (outspoken) wheels, n -ary trees of different levels etc.Item Set colorings of digraphs(Utilitas Mathematica Publishing Inc., 2016) Hegde, S.M.; Castelino, L.P.A set coloring of the digraph D is an assignment (function) of distinct subsets of a finite set X of colors to the vertices of the digraph, where the color of an arc, say (u, v) is obtained by applying the set difference from the set assigned to the vertex v to the set assigned to the vertex u which are also distinct. a set coloring is called a strong set coloring if sets on the vertices and arcs are distinct and together form the set of all non empty subsets of X. a set coloring is called a proper set coloring if all the non empty subsets of X are obtained on the arcs. a digraph is called a strongly set colorable (properly set colorable) if it admits a strong set coloring (proper set coloring). In this paper we give some necessary conditions for a digraph to admit a strong set coloring (proper set coloring), characterize strongly (proper) set colorable digraphs such as directed stars, directed bistars etc.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.