Research On Problems Related To Rainbow Number

An edge-colored graph is said to be rainbow if every edge of the graph has a different color.The rainbow number of a graph H in a graph G,denoted by rb(G,H),is the smallest integer k such that any edge coloring of graph G has a rainbow subgraph which is isomorphic to graph H.The rainbow number is closely related to anti-Ramsey numbers.In recent years,researchers from various countries have carried out extensive research on the rainbow number of different graph G and H.In this paper,we call a complete bipartite graph with arbitrary one edge removed as an edge-removed complete bipartite graph,denoted by Km,n-.The rainbow number rb(G,kK2)of mathching in the graph G is called the rainbow matching number of the graph G.This paper mainly studies the rainbow matching number of the edgeremoved complete bipartite graph,and compares it with the rainbow matching number of the complete bipartite graph in addition,this paper also studies whether the rainbow matching contained in a complete bipartite graph is unique after coloring with the rainbow matching number.The main results of this paper are as follows:1.For all positive integers m,n,k,we get the exact value of the rainbow matching number of the edge-removed complete bipartite graph Km,n-,and compare with the rainbow matching number of the complete bipartite graph.2.For all positive integers m,n,k,we get all cases such that after the complete bipartite graph is coloring with its rainbow matching number,the rainbow matching contained in the graph is unique,and it is proved that other cases are not unique.