Research On The Graceful Labeling Of Directed Graph 2k(?)

The problem of labeling graphs is a research topic with practical application background in graph theory.In recent years,domestic and foreign research in this area has been relatively active,and the research results have been applied to radio astronomy,X-ray diffraction crystallography,cryptography and reading DNA sequences etc.Its research began in 1966.A.Rosa's famous graceful tree conjecture.The vertex labels of a graph is the mapping of the graph's vertex set to the integer set(which can also be a commutative group),and the edge label is the mapping of the graph's edge set to the integer set.Due to the continuous emergence of new application backgrounds,some new labels and public problems have arisen,the construction methods and theories of these labels need to be further studied.These studies have important theoretical significance and application value.In the book "Graceful Graph",Ma kejie introduced several types of research on the gracefulness of directed graphs,and proposed several open questions and conjectures for continued research,among them,the conjecture "When n is even,the directed graph n(?)3 are graceful graph",until now this has not been completely resolved,we studied the conjecture.This article is divided into three chapters.The first chapter introduces some basic concepts of directed graphs and hypergraphs.In the second chapter,the differential mode method and the theory of completely 3-uniform hypergraphs are used to study the gracefulness of directed graph n(?)3,we obtain that when n?0(mod 2)and 2?n?16,the directed graph n(?)3 is graceful.In the third chapter,we continue to study the gracefulness of (?)3,and obtain the feasible graceful edge label of the directed graph n(?)3 when n?0(mod 4)and 64?n?94.