The Problems Of Disjoint Cycles In Digraphs

In the study of directed graphs,the existence of disjoint cycles has always been one of the key contents of the study,and has very important research significance,but its research is very difficult.In recent years,many researchers have begun to investi-gate whether a directed graph contains two disjoint cycles with different lengths.In particular,Tan proved in 2015 that every 3-regular bipartite digraph with a cycle fac-tor of at least two cycles contains two disjoint cycles with different lengths.Moreover,Tan put forward a conjecture in 2020:every 3-regular digraph,except the digraphs D₇~3,D₈~3and D~2_2n(n?2),contains two disjoint cycles of different lengths.Based on this,this paper mainly studies the related problems of disjoint cycles in 3-regular digraphs.It is divided into four chapters.The specific contents are as follows:The first chapter introduces some symbols and definitions used in this paper,and summarizes the research progress of disjoint cycles in digraphs.In Chapter 2,we extend the 3-regular bipartite digraph from Tan's conclusion in2015 to 3-regular digraph,and study it.Furthermore,we prove that every 3-regular digraph,which possesses a cycle factor with at least three cycles,contains two disjoint cycles of different lengths,except for the digraph that is isomorphic to the D~2_2n(n?3).In Chapter 3,we give a sufficient condition for a 3-regular digraph with cycle factor of two cycles to contain two disjoint cycles with different lengths.In addition,we also draw several inferences under some restrictive conditions.In the fourth chapter,we summarize the research content of this paper,and put forward some problems that can be further studied.