The Eulerian Cover Number Of Cartesian Product Digraphs

In 1736,Euler studied the problem of seven bridges in Konigsberg.He not only solved the seven-bridge problem,but also opened up a new branch of mathematics.With the birth of the world's first computer on February 14,1946,Graph theory enjoyed a rapid development and has been widely applied in transportation,computer science and other disciplines.In fact,along with the rise of computer technology,Graph theory has grown beyond itself.With computer technology,many difficult problems in Graph theory were tackled;moreover,the theory got used in various walks of life,successfully combining theory and practice.In 1977,Boesch,Suffel,Tindell et al.first presented the supereulerian problem before the public based on the postman problem in China.This paper only involves the supereulerian problem in Graph theory.The supereulerian property of digraphs has been an important and novel research direction of the Graph theory,for which it has also become a hot issue in the research of the Graph theoryThe text of the paper consists of three chapters:In the first chapter,firstly,we introduce the research background and status quo and basic concepts needed of supereulerian digraph,then,give definitions of cartesian product digraph.Finally,we introduce contents and main result in this paperIn the second chapter,according to the definition of cartesian product digraph,we obtain the property of cartesian product digraph;secondly,We mainly studied the eulerian cover number of the digraph obtained by deleting any points of the cartesian product of two supereulerian digraph.Moreover,we characterize digraphs with points less than 4 and euler coverage of 2.Finally,we obtain the eulerian cover number of the cartesian product of strong digraph D1 with ec(D1)=land the digraph D2 with ec(D2)=2In the third chapter,we first show the eulerian cover number of the digraph obtained by deleting any edges of the cartesian product of two supereulerian digraph.Finally,we mainly research the eulerian cover number of the cartesian product of strong digraph D1 with ec(D1)=1 and the digraph D2 with ec(D2)=3.