Browsing by Author "Shivarajkumar"

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Further Results on Graceful Digraphs
(2016) Hegde, S.M.; Shivarajkumar
A 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.
Further Results on Graceful Digraphs
(Springer, 2016) Hegde, S.M.; Shivarajkumar
A 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.
Graceful labeling of digraphs—a survey
(Taylor and Francis Ltd., 2021) Shivarajkumar; Sriraj, M.A.; Hegde, S.M.
A digraph D with p vertices and q arcs is labeled by assigning a distinct integer value g(v) from (Formula presented.) 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)) (mod q + 1) If the arc values are all distinct then the labeling is called a graceful labeling of digraph. In this survey article, we have collected results that we could find interesting on graceful labeling of digraphs. © 2021 The Author(s). Published with license by Taylor & Francis Group, LLC.
On k-graceful digraphs
(2014) Hegde, S.M.; Shivarajkumar
In this paper we extend the idea of k-graceful labeling of undirected graphs to a directed graphs: A simple directed graph D with n vertices and e edges is labeled by assigning each vertex a distinct element from the set ?c+k = {0,1,2.....e + k - 1}, where is a positive integer and an edge xy from vertex x to vertex y is labeled with ?(x, y) = ?(y) - ?(x)mod(e + k), where ?(y) and ?(x) are the values assigned to the vertices y and x respectively. A labeling is a k-graceful labeling if all ?(x, y) are distinct and belong to {k, k + 1,k + e-1}. If a digraph D admits a k-graceful labeling then D is a fc - graceful digraph. We also provide a list of values of fc for which the unidirectional cycle C?n admits a k-graceful labeling. Further, we give a necessary and sufficient condition for the outspoken unicyclic wheel to be k-graceful and prove that to provide a list of values of k > 1, for which the unicyclic wheel W?n is fc-graceful is NP - complete.
On k-graceful digraphs
(Utilitas Mathematica Publishing Inc., 2014) Hegde, S.M.; Shivarajkumar
In this paper we extend the idea of k-graceful labeling of undirected graphs to a directed graphs: A simple directed graph D with n vertices and e edges is labeled by assigning each vertex a distinct element from the set ?c+k = {0,1,2.....e + k - 1}, where is a positive integer and an edge xy from vertex x to vertex y is labeled with ?(x, y) = ?(y) - ?(x)mod(e + k), where ?(y) and ?(x) are the values assigned to the vertices y and x respectively. A labeling is a k-graceful labeling if all ?(x, y) are distinct and belong to {k, k + 1,k + e-1}. If a digraph D admits a k-graceful labeling then D is a fc - graceful digraph. We also provide a list of values of fc for which the unidirectional cycle C?n admits a k-graceful labeling. Further, we give a necessary and sufficient condition for the outspoken unicyclic wheel to be k-graceful and prove that to provide a list of values of k > 1, for which the unicyclic wheel W?n is fc-graceful is NP - complete.
Two Conjectures on Graceful Digraphs
(2013) Hegde, S.M.; Shivarajkumar
A 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))(mod q + 1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. Bloom and Hsu (SIAM J Alg Discr Methods 6:519-536, 1985) conjectured that, all unicyclic wheels are graceful. Also, Zhao et al. (J Prime Res Math 4:118-126, 2008) conjectured that, for any positive even n and any integer m ? 14, the digraph n-{long rightwards arrow}Cm} is graceful. In this paper, we prove both the conjectures. � 2012 Springer.
Two Conjectures on Graceful Digraphs
(2013) Hegde, S.M.; Shivarajkumar
A 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))(mod q + 1). If the arc values are all distinct then the labeling is called a graceful labeling of a digraph. Bloom and Hsu (SIAM J Alg Discr Methods 6:519-536, 1985) conjectured that, all unicyclic wheels are graceful. Also, Zhao et al. (J Prime Res Math 4:118-126, 2008) conjectured that, for any positive even n and any integer m ? 14, the digraph n-{long rightwards arrow}Cm} is graceful. In this paper, we prove both the conjectures. © 2012 Springer.