An application of Turán theorem to domination in graphs ∗

3 A function f : V (G) → {+1,−1} defined on the vertices of a graph G is a signed domi4 nating function if for any vertex v the sum of function values over its closed neighborhood 5 is at least one. The signed domination number γs(G) of G is the minimum weight of a 6 signed dominating function on G. By simply changing “{+1,−1}” in the above definition 7 to “{+1, 0,−1}”, we can define the minus dominating function and the minus domination 8 number of G. In this note by applying Turán theorem we present sharp lower bounds on 9 the signed domination number for a graph containing no (k + 1)-cliques. As a result, we 10 generalize a previous result due to Kang et al. on the minus domination number of k-partite 11 graphs to graphs containing no (k + 1)-cliques and characterize the extremal graphs. 12 AMS (2000) subject classification: 05C35, 05C69 13