Three conjectures on the signed cycle domination in graphs

Abstract

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 ∑e∈E(C)g(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as γsc(G)=min{∑e∈E(G)g(e)∣g 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 n≥3. Then γsc(G)=n−2; (2) For any graph G with δ(G)=3, γsc(G)≥1; (3) For any 2-connected graph G, γsc(G)≥1. In this paper, we present some results about these conjectures.