Cyclically Interval Total Colorings Of Some Join Graphs

An interval is a set of consecutive integers.For any two integers a and b with a≤b,we denote by[a,b]the interval {a,a+1,…,b-1,6}.If a>b,then[a,b]=?.For any integers a,b and t with 1<a≤b<t,[1,t]\[a,b]is called a cyclic interval.If M is an interval or a cyclic interval with |M|=k,then M is called a k-(cyclic)interval.For a total t-coloring α of a graph G and for any vertex v∈V(G),if S[α,v]is a(dG(v)+1)-interval or a(dG(v)+1)-cyclic interval,then a is called a cyclically interval total t-coloring of G,and G is called cyclically interval t-total colorable(or:cyclically interval total colorable),where S[α,v]is the set {α(v)}∪{α(e)|e is incident to v},and dG(v)is the degree of v in G.The set of all cyclically interval total colorable graphs is denoted by(?).Let Θ(G)={t|G is cyclically interval t-total colorable},ωτc(G)=min Θ(G)and Wτc(G)=max Θ(G).By starting with a disjoint union of two graphs G and H and adding edges joining every vertex of G to every vertex of H,one obtains the join of G and H,denoted G∨H.In this paper,we study the cyclically interval total coloring of the complete bipartite graph Km,n,the join Im∨Cn of an empty graph Lm and a cycle Cn,the join Pm∨Pn of a path Pm and another path Pn,the join Cm∨Cn of a cycle Cm and another cycle Cn,and the join of a path Pm and acycle Cn.1.For any m,n∈N,we proved that Km,n∈F,and got the following results.(1)(?)(2)Wτc(Km,n)≥m+n+4,where min{m,n}>1.2.For any m,n∈N,we showed that if n=m+1,m+2 and m≥2 is even,or m≥n≥3,then ωτc(Im∨Cn)=m+3;and if m≥2,n≥3,then Wτc(Im∨Cn)≥m+n+4.3.For any integers m,n≥2,we proved that Pm∨Pn∈F,and got some results as follow:(1)(?)(2)n+3≤ωτc(Pn∨Pn)≤n+4,where n≥3,(3)ωτc(Pm∨Pn)=n+3,where n>m≥3,(4)(?)4.For any integers m,n≥3,we showed that,if n>m and m is even,or n=m+1,m+2 and m is odd,or n≥2m+2 and m is odd,then Cm∨Cn∈F and ωτc(Cm∨Cn)=n+3.5.For any integers m,n≥2,we proved that Pm∨Cn∈F,and got the following results.(1)ωτc(P2∨Cn)=n+2,where n≥2,(2)有n+3≤ωτc(Pn∨Cn)≤n+4,where n≥3,(3)ωτc(Pm∨Cn)=max{m,n}+3,where n≠m and m≥3.