For a positive integer k,the total{k}-dominating function(T{k}DF)of a graph G without isolated vertices is a function f from the vertex set V(G)to the set{0,1,2,…,k}such that for each vertex v∈V(G),the sum of the va...For a positive integer k,the total{k}-dominating function(T{k}DF)of a graph G without isolated vertices is a function f from the vertex set V(G)to the set{0,1,2,…,k}such that for each vertex v∈V(G),the sum of the values of all its neighbors assigned by f is at least k.A set{f_(1),f_(2),…,f_(d)}of pairwise different T{k}DF s of G with the property that∑d i=1 f_(i)(v)≤k for each v∈V(G),is called a total{k}-dominating family(T{k}D family)of G.The total{k}-domatic number of a graph G,denoted by d^({k})_(t)(G),is the maximum number of functions in T{k}D family.In 2013,Aram et al.proposed a problem that whether or not d^({k})_(t)(C_(m)□C_(n))=3 when 4 nmk,and d^({k})_(t)(C m□C n)=4 when 4|nmk.It was shown that d^({k})_(t)(C_(m)□C_(n))=3 if 4 nmk and k≥2 or 4|nmk and 2 nk,which partially answered the above problem.In addition,the total{k}-domatic number of the direct product of a cycle and a path,two paths,and two cycles was studied,respectively.展开更多
基金Supported by NNSF of China(11671376,11401004)Anhui Provincial Natural Science Foundation(1708085MA18)
文摘For a positive integer k,the total{k}-dominating function(T{k}DF)of a graph G without isolated vertices is a function f from the vertex set V(G)to the set{0,1,2,…,k}such that for each vertex v∈V(G),the sum of the values of all its neighbors assigned by f is at least k.A set{f_(1),f_(2),…,f_(d)}of pairwise different T{k}DF s of G with the property that∑d i=1 f_(i)(v)≤k for each v∈V(G),is called a total{k}-dominating family(T{k}D family)of G.The total{k}-domatic number of a graph G,denoted by d^({k})_(t)(G),is the maximum number of functions in T{k}D family.In 2013,Aram et al.proposed a problem that whether or not d^({k})_(t)(C_(m)□C_(n))=3 when 4 nmk,and d^({k})_(t)(C m□C n)=4 when 4|nmk.It was shown that d^({k})_(t)(C_(m)□C_(n))=3 if 4 nmk and k≥2 or 4|nmk and 2 nk,which partially answered the above problem.In addition,the total{k}-domatic number of the direct product of a cycle and a path,two paths,and two cycles was studied,respectively.
