(Adjacent) Vertex Distinguition Coloring Problem Research Of Special Graphs

In this paper,we studied the(adjacent)vertex distinguition equitable edge(total)coloring of the roads and stars join-gruph,and the adjacent Vertex distinguishable total coloring of the crown graph Gn.m as well as vertex-distinguishing equitable edge coloring of product graphs between complete graph and the complete graph,star and star,wheel and wheel.1.For the join-graph Pn ? Sn of two finite graph Pn and Sn,we obtained the equitable adjacent strong edge chromatic number of Pn ? Sn.Specifically,when m=1,for P1 ? Sn,then Xeas'(P1?Sn)=n+2.when m=2,for P2 ? Sn,then Xeas(P2?Sn)=?when m=3,for P3?Sn,then Xeas'(P3?Sn)=n+4.when n?4,for Pn?Sn,then Xeas'(Pn?Sn)=2n.2.For the join-graph Pm ? Sn of two finite graph Pm and Sn,we obtained the adjacent vertex-dinguishing equitable total chromatic number of Pn?Sn.Of Pn?Sn at m=1,2,3,n?1 and m=n?4.Specifically,when m=1,for P1?Sn,then Xeat(P1?Sn)=?when m=2,for P2?Sn,then Xaet(P2?Sn)=n+4.when m=3,for P3?Sn,then Xaet(P3?Sn)=n+5.3.For the join-graph Pm?Sn of two finite graph Pm and Sn,we obtained the vertex-dinguishing equitable edge chromatic number of Pn?Sn at m=1,2,3,n?1 And m=n?4.Specifically,when m=1,for P1?Sn,then Xvde'(P1?Sn)=n+2.when m=2,for P2?Sn,then Xvde'(P2?Sn)=? when m?3,for P3?Sn,then Xvde'(P3?Sn)=n+4.when m=n?,for Pn?Sn,Shen Xvde'(Pn?Sn)=2n.4.For the join-graph Pm?Sn of two finite graph Pm and Sn,we obtained the vertex-dinguishing equitable total chromatic number of Pn?Sn of Pn?Sn at m=1,2,3,n?1.Specifically,when m=1,for P1?Sn,then Xve(P1?Sn)=?when m=2,for P2?Sn,then xvet(P2?Sn)=n+4.when m=3,for P3?Sn,then Xvet(P3?Sn)=n+5.5.For the vertex-dinguishing equitable edge coloring of Cartesian graph,we obtained the vertex-dinguishing equitable edge chromatic numbers of Cartesian graphs between complete graph and complete graph,star and star,wheel and wheel,which verify the conjecture on VDEECC.Specifically,(1).let G1 and G2 are two simple graph,if Xvde'(Gi)=?(Gi),i=1,2.|E(Gi)|?0(mod ?(Gi)),i=1,2.|E(G1)|·|V(G2)|/?(G1)=|E(G2)|·|V(G1)|/?(G2)then for Cartesian graph between G1 and G2,Xvde'(G1?G2)=Xvd',(G1?G2)=?(G1)+?(G2).(2).let Kn?Kn' be the Cartesian graph between complete graph Kn and Kn',if n?2,then Xvde'(Kn?Kn')=Xvd'(Kn?Kn')=2n.(3).Let G1 and G2 are two simple graph,if Xvde'(G1)=?(G1),Xvde'(G2)=?(G2)+1,and the max-degree vertex of G2 i s not unique;|E(G1)|=0(mod ?(G1)),|E(G2)|?0(mod A(G2)+1);|E(G1)|·|V(G2)|/?(G1)=|E(G2)|·|V(G1)|/?(G2)+1.then for the for Cartesian graph between G1 and G2,Xvde'(G1?G2)=Xvd\(G1?G2)=?(G1)+?(G2)+1.6.For the adjacent vertex-dinguishing total coloring of the corwn graph Gn,m,and its adjacent vertex dinguishable total chromatic number is obtained.Specifically,if Gn,m(n?3,m?1),then Xat(Gn,m)=m+5.