The Adjacent Vertex Distinguishing Colouring Of Digraphs

As an important branch of combinatorial mathematics,graph theory is widely used in management science,computer science and technology,communication engineering and other fields.In the research process of graph theory,the coloring problem of graphs occupies an important position.The coloring of the graph refers to the classification of vertices,edges and other elements in the graph according to certain rules.According to the different coloring objects and coloring rules,different types of coloring concepts are generated,such as vertex coloring,edge coloring,face coloring and full coloring,among which the most studied and perfect results is the edge coloring.The edge coloring of digraphs is a new problem raised in recent years,which is closely related to complex networks,bioinformatics and other fields,and has important theoretical value and application background.In the study of non-regular networks,it is sometimes necessary to assign a positive integer weight to the edges of the graph,so that the edge weights associated with each vertex in the graph are different.Similar practical problems promote the development of the coloring theory of the graph.This paper mainly studies the adjacent vertex distinguishing coloring of the digraph.The main work is as follows.First of all,the(-,(10))variant of adjacent vertex distinguishing coloring of the digraph is studied.Under the condition of the(-,(10))variant,for directed graphs without lonely arcs and ss arcs,the best result of the adjacent vertex distinguishing chromatic number is less than or equal to 3.It is known that it is meaningful to continue to study adjacent vertex distinguishing chromatic number of the digraph does not exceed 2,so starting from some special digraph,study their adjacent vertex distinguishing chromatic number.It is proved that for the directed path,directed circle,the join of directed circle and directed circle,the product of directed path and directed circle,and the Cartesian product of directed path and directed circle,their adjacent vertex distinguishing chromatic number is equal to 2.Secondly,the((10),(10))variant of adjacent vertex distinguishing coloring of the digraph is studied.It is known that the best result of the((10),(10))variant is: for any digraph,adjacent vertex distinguishing chromatic number is less than or equal to 3.It has been proved that for the directed path,directed circle,the join of directed circle and directed circle,the product of directed path and directed circle,the Cartesian product of directed path and directed circle and the lexicographic product graphs of directed path and directed circle,their adjacent vertex distinguishing chromatic number of((10),(10))variant is equal to 2.Finally,it is studied that adjacent vertex distinguishing proper arc coloring of the digraph.On the basis of adjacent vertex distinguishing arc coloring of the digraph,it is required that coloring of the digraph is proper.Based on the equivalence between adjacent vertex distinguishing proper arc coloring of the digraph and the neighbor vertex distinguishing edge coloring of bipartite graph,a new coloring conjecture is proposed,and the problem of adjacent vertex distinguishing proper arc coloring of complete digraph and the directed tree is explored.