Research On The K-proper Index Of Split Graphs

Let(G.c)be an edge-colored graph and let T be a tree of G.T is called a proper tree of(G,c)if no two adjacent edges of T are assigned the same color under c.For a vertex subset S of G,a tree T of G is called an S-tree if S(?)V(T).Let k be a fixed integer with k≥2.A k-proper coloring of G is defined to be an edge coloring c of G which has the property that,for every subset S ol k vertices of G.there is a proper S-tree in(G,c).The minimum number of colors that are required in a k-proper coloring of G is called the k-proper index of,(denoted by pxk(G).(m,n)-split graph,denoted by Sm,n,is a typical network structure.The vertex set of(m,n)-split graph is given by V=X ∩Y,where X is a clique and Y is an independent set,and moreover,every vertex in Y is adjacent to every vertex in X.Based on the importance of k-proper index of graphs in communication network and the neat composite structure of(m,n)-split graph,we study the k-proper index of(m,n)-sppit graph in this thesis.The main results of this paper are as follows:(1)If m+n≥3 and m≥2,then px2(Sm,n)=2.(2)If n≥2,then pxk(S1,n)=n for each integer k with 3≤k≤n+1.(3)If m+n≥3 and m≥n-1,then pxk(Sm,n)=2 for each integer k with 3≤k≤m+n.(5)If 3≤m≤n-2,then 2≤px3(Sm,n)≤3.(7)If 4≤m≤n-2,then pa3(Sm,n)=2.(8)If 4≤k≤m≤n-2,then 2≤pxk(Sm,n)≤3.(9)If 2(fk-1)≤m≤n-2 and k≥4,then pxk(Sm,n)=2。...