On the dimension of vertex labeling of $k$ -uniform dcsl of an even cycle

On the dimension of vertex labeling of $k$ -uniform dcsl of an even cycle

In this paper, we discuss the lower bound for the dcsl index δk of a k -uniform dcsl of even cycle $C_{2n}$, n ≥ 2 , in terms of the dimension of a poset and prove that dim($mathscr{F}$ ) ≤ δk (C2n) , where $mathscr{F}$ is the range of any k -uniform dcsl $f$ of $C_{2n}$, n ≥ 2 .

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  • 1] Acharya BD. Set-valuations of graphs and their applications. MRI Lecture Notes in Applied Mathematics 1983; 2: 1-25.
  • [2] Acharya BD, Germina KA. Distance compatible set-labeling of graphs. Indian Journal of Mathematics and Com- puter Science 2011; 1: 49-54.
  • [3] Baker K, Fishburn P, Roberts F. Partial orders of dimension 2, interval orders and interval graphs. Networks 1971; 2: 11-28.
  • [4] Birkhoff G. Lattice Theory. Providence, RI, USA: AMS, 1967.
  • [5] Bogart KP. Maximal dimensional partially ordered sets I Hiraguchi’s theorem. Discrete Math 1973; 5: 21-31.
  • [6] Dilworth RP. A decomposition theorem for partially ordered sets. Ann Math 1950; 51: 161-166.
  • [7] Djokovic DZ. Distance preserving subgraphs of hypercubes. J Combin Theory Ser B 1973; 14: 263-267.
  • [8] Doignon JP, Falmagne JCl. Well graded families of relations. Discrete Math 1997; 173: 35-44.
  • [9] Dushnik B, Miller EW. Partially ordered sets. Am J Math 1941; 63: 600-610.
  • [10] Eppstein D, Falmagne JCl, Ovchinnikov S. Media theory. Discrete Applied Mathematics 2002; 121: 83-101.
  • [11] Falmagne JCl. Stochastic token theory. J Math Psychol 1997; 41: 129-143.
  • [12] Felsner S, Trotter WT, Wiechert V. The dimension of posets with planar cover graphs. Graph Combinator 2014; 1-13.
  • [13] Germina KA. Uniform distance-compatible set-labelings of graphs. Journal of Combinatorics, Information and System Sciences 2012; 37: 169-178.
  • [14] Germina KA, Thomas BK. Distance compatible set-labeling index of graphs. Int J Contemp Math Sciences 2010; 5: 911-919.
  • [15] Germina KA, James J. Characterization of 1-uniform dcsl graphs using well-graded family of sets. Advances and Applications in Discrete Mathematics 2015; 15: 113-123.
  • [16] Germina KA, Nageswararao K. Characterization of vertex labeling of 1-uniform dcsl graph which form a lattice. Journal of Fuzzy Set Valued Analysis 2015; 2015: 166-170.
  • [17] Harary F. Graph Theory. Boston, MA, United States: Addison-Wesley, 1969.
  • [18] Hiraguchi T. On the dimension of orders. Sci Rep Kanazawa Univ 1955; 4: 1-20.
  • [19] Kelly D. On the dimension of partially ordered sets. Discrete Math 1981; 35: 135-156.
  • [20] Mirsky L. A dual of Dilworth’s decomposition theorem. Am Math Mon 1971; 78: 876-877.
  • [21] Nageswara Rao K, Germina KA, Shaini P. On the dimension of vertex labeling of k-uniform dcsl of k-uniform caterpillar. Carpathian Math Publ 2016; 8: 134-149.
  • [22] Ovchinnikov S. Partial cubes: Structures, characterizations, and constructions. Discrete Math 2008; 308: 5597-5621.
  • [23] Trotter WT. Dimension of the crown $S^k_n$ . Discrete Math 1974; 8: 85-103.
  • [24] Trotter WT, Moore J. The dimension of planar posets. J Combin Theory B 1977; 22: 54-67.
  • [25] Winkler PM. Isometric embedding in products of complete graphs. Discrete Appl Math 1984; 7: 221-225.