On The Paths And Cycles Of Graphs

The research of paths and cycles of graphs is a topic in Graphs Theory which is important and has been thrived for a long time. Because many scholars had worked on the paths and cycles for many years, we have some good results on them.Recently, researches on the paths and cycles are mainly gone on in some special fields. In this paper, we try to find and prove some characters on the paths and cycles in simple graphs, in digraphs and in colored graphs. Some satisfied results are gained.In first chapter, on the paths and cycles in simple graphs, we try to find some problems from different new ways and resolve them, thus, the theories on the paths and cycles of simple graphs are more fruitful.In second chapter, we go on some researches about the shortest cycles in digraphs and about C-H conjecture. At first, we look back on the development of C-H conjecture. Second, we discuss several necessary conditions for possible counterexamples to C-H conjecture, these conditions have not enable us to prove or refute the conjecture, but they lead to proofs of special instances of the conjecture. Finally, we go on some basic researches about the problems in strongly connected digraphs which are relative to C-H conjecture and some basic researches about the directed cycles in hyper-digraphs.In third chapter, we go on some deep researches on the paths and cycles in colored graphs. For easily doing so, we go on from four cases. [That is: the normal vertex-coloring, the normal edge-coloring, the common vertex-coloring, the common edge-coloring]. In each case, we mainly research the characters of paths and cycles: the lengths of paths and cycles are required as long as possible, on the other hand, the different colors in the paths and cycles are required as many as possible, hi the best case, we may find a longest path (or cycle) which contains most colors.