The Connectivity Of Edge-Colored Graphs

As the origin of graph coloring problem,the four-color conjecture has always led the development of graph theory,and made graph coloring theory an important branch of graph theory.Connectivity is one of the most important properties of graph theory.Based on these aspect.s,the connectivity problem of colored graphs emerges at the historic moment.An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured.The properconnectionnumber of a connected graph G,denoted by pc(G),is the smallest number of colours that are needed in order to make G properly connected.For each pair of positive integers n and k,we define by g(n,k)the smallest integer such that every connected graph of order n and size at least g(n,k)has proper connection number at most k.In 2016,Susan A.van Aardt et al.completely determine the function g(n,k).Based on this,by adding minimum degree condition consider the relationship between proper connection number and size.In addition,because hypercubes have abundant symmetry and many applications in net-work design and reliability,we give priority to hypercubes for connectivity of special chromatic graphs.In 2017,Eddie Cheng et al.defined(j)-coloring(1 ?j<n)in the hypercube Hn.On this basis,we studied The proper distance between any two vertices is studied and when j = 1,the number of the shortest normal coloring paths is different between any two vertices.This paper further extends this conclusion and studies the number of the shortest normal coloring paths which are different between any two vertices of j.