Research On Two Types Of Domination Numbers Of Product Graphs

Graph theory is a very applicable mathematical discipline and its application is becoming more and more widespread with the development of the era of big data information.Graph dominate theory is an important research direction of graph theory,which is also widely used in real life and other disciplines.In this paper,we study the weak 2 domination number of the Kronecker product graph of paths Pn × Pm and the k-rainbow domination number(4?k?7)of the Cartesian product graph of cycles Cn?Cm.For the weak 2 domination number of the Kronecker product graph Pn×Pm,this paper uses computer construction and mathematical deduction to prove the values or bounds.First,the weak 2 dominating function is constructed according to the graphical characteristics,and the upper bounds of the weak 2 domination number are obtained based on the function.Then,we use mathematical derivation to prove the lower bound of the weak 2 domination number of Pn×P3.Finally,the exact value of the weak 2 domination number of Pn×P3 and the bounds of other cases of Pn×Pm are obtained.For k-rainbow domination numbers of Cartesian product graphs Cn?Cm,this paper first studies the 4-rainbow domination number.We obtained some tight upper bounds by constructing good dominating functions,so that parts of upper bounds are equal to the known lower bounds,thereby we can obtain some exact values and some upper and lower bounds in other cases.Then,the 4-Rainbow dominating is extended to k-rainbow dominating(5 ?k?7),and the exact value of the k-rainbow domination number is obtained when n?0(mod 4)and m?0(mod 4).