摘要
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of distinct Roman k-dominating functions on G with the property that ∑di=1 fi（v） ≤ 2 for each v C V（G）, is called a Roman k-dominating family （of functions） on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by dkR（G）. Note that the Roman 1-domatic number dlR（G） is the usual Roman domatic number dR（G）. In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for dkR（G）. In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
Let k be a positive integer. A Roman k-dominating function on a graph G is a labeling f ： V（G） → {0, 1, 2} such that every vertex with label 0 has at least k neighbors with label 2. A set {f1, f2,..., fd} of distinct Roman k-dominating functions on G with the property that ∑di=1 fi（v） ≤ 2 for each v C V（G）, is called a Roman k-dominating family （of functions） on G. The maximum number of functions in a Roman k-dominating family on G is the Roman k-domatic number of G, denoted by dkR（G）. Note that the Roman 1-domatic number dlR（G） is the usual Roman domatic number dR（G）. In this paper we initiate the study of the Roman k-domatic number in graphs and we present sharp bounds for dkR（G）. In addition, we determine the Roman k-domatic number of some graphs. Some of our results extend those given by Sheikholeslami and Volkmann in 2010 for the Roman domatic number.
