I_(m)∨C_(n)的循环区间全着色

Cyclically Interval Total Colorings of I_(m)∨C_(n)

k-区间就是由k个连续整数构成的集合.对于图G的t-全着色α以及任意顶点v∈V(G),如果S[α,v]为[d_(G)(v)+1]-区间,或者{1,2,…,t}\S[α,v]为[t-d_(G)(v)-1]-区间,则称α为G的循环区间t-全着色,并称G为可循环区间全着色的,其中S[α,v]表示{α[v]}∪{α[e]|e与v关联},dG(v)表示顶点v在图G中的度.所有可循环区间全着色的图构成的集合记作F.对于任意图G∈F,其循环区间全着色所需最少颜色数记作w_(τ)^(c)(G).研究空图I_(m)与圈C_(n)的联图I_(m)∨C_(n)(m≥2,n≥3)的循环区间全着色,证明I_(m)∨C_(n)∈F,并且除了个别情况(n=m+2且m≥2为奇数),得到了w_(τ)^(c)(I_(m)∨C_(n))的准确值.

A k-interval is a set of k consecutive integers.For a total t-coloringα of a graph G and for any vertex v∈V(G),if S[α,v]is a[d_(G)(v)+1]-interval,or{1,2,…,t}\S[α,v]is a[t-d_(G)(v)-1]-interval,thenαis called a cyclically interval total t-coloring of G,and G is called cyclically interval total colorable,where S[α,v]is the set{α[v]}∪{α[e]|e is incident to v},and d_(G)(v)is the degree of v in G.The set of all cyclically interval total colorable graphs is denoted by F.For any graph G∈F,the least number of colors used in a cyclically interval total coloring of G is denoted by w_(τ)^(c)(G).The cyclically interval total coloring of the join Im∨Cn(m≥2,n≥3)of an empty graph I_(m) and a cycle C_(n) is studied.It proves that I_(m)∨C_(n)∈F,and except for some cases(n=m+2 and m≥2 is odd),also get the exact value of w_(τ)^(c)(Im∨Cn)