Three conjectures on the signed cycle domination in graphs

Let G=(V,E) be a graph, a function g:E ->{-1,1} is said to be a signed cycle dominating function (SCDF for short) of G if a (eaE(C)) g(e)a parts per thousand yen1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as gamma (sc) (G)=min{a (eaE(G)) g(e)a g pound is an SCDF of G}. Xu (Discrete Math. 309:1007-1012, 2009) first researched the signed cycle domination number of graphs and raised the following conjectures: (1) Let G be a maximal planar graphs of order na parts per thousand yen3. Then gamma (sc) (G)=n-2; (2) For any graph G with delta(G)=3, gamma (sc) (G)a parts per thousand yen1; (3) For any 2-connected graph G, gamma (sc) (G)a parts per thousand yen1. In this paper, we present some results about these conjectures.