Four Labellings Of Trees With Trees Of Order12and Edges Joining Trees

Graph labelling problems can be traced Rosa’s famous graceful tree conjectureposed in1967.A vertex labelling is a mapping from the vertex set of a graph toa set of integers, and edge labelling is a mapping from the edge set of a graph toa set of integers. According to the diferent requirements of mappings on vertexsets or edge sets of graphs, there are many interesting graph labellings. It canbe applied to a wide of scientifc areas and other practical felds. In this paperour investigations are related with four labellings of some particucal trees and thetrees of order12:(k, d)-graceful labellings,(k, d)-strong graceful labellings,(k,d)-odd graceful labellings and odd-graceful labellings. The main results are asfollows:(1) Introduced (k, d)-graceful labellings,(k, d)-strong graceful labellings,(k, d)-odd graceful labellings of some particucal trees.(2) Using the enumeratingmethod produces all non-isomorphic trees of order12. Furthermore, we show theodd-gracefulness of trees of order12.(3) We show the strongly graceful labellingsof trees of order12and fnd (k, d)-graceful labellings of a large proportion of treesof order12.