The Rainbow Connection Number Of Several Kinds Of Graphs Related To Cactus Graphs And The Edge Metric Dimension Of Cactus Graphs

The concept of rainbow connection was introduced by G.Chartrand et al in 2008.Compute the rainbow connection number of connected graphs is NP-hard,so it is mean-ingful to calculate the rainbow connectivity number of a specific graph.The first part of this paper studies the rainbow connection number of some kinds of graphs related to the corona graphs and the rainbow connection number of cartesian product graphs.In 1976,F.Harary and R.A.Melter independently introduced the concept of metric dimensions for graphs;In 2016,A.Kelenc et al proposed the concept of the edge metric dimension.The cactus graph consists of cut edges or circles.The second part of this paper is to solve the edge metric dimension of cactus graphs by the edge metric dimension of the circle obtained at present.The conclusion is as f'ollows:1.An upper bound on the rainbow connection number of corona graphs G?K1 is n+k+l,where G is a cactus graph.An upper bound on the rainbow connection number of corona graphs G ? K2 is k+3l+n-?il=1ni/2,where G is a cactus graph,where i=1,2,…l.2.An upper bound on the rainbow connection number of cartesian product graph G?K2 is?ni+?nj-s+l+k-h,where G is a cactus graph.An upper bound on the rainbow connection number of the cartesian product G?Pn is(n-1)(?ni+?nj-s-h)+l+k,where G is a cactus graph,where i=1,2?…l;j=l+,l+2,…,l+m.3.An upper bound on the edge metric dimension of cactus graphs is k+h+m+s+2l+p+2q+w+2.