On a Conjecture of Cockayne E J etc.
Baogen Xu · Journal of Nanchang University · 2006
Cockayne E J introduced the concept of the signed k-subdomination number γ~(-11)_(ks)(G) of a graph G,posed a conjecture as follows:For any connected graph G of order n and integer k(n2k≤n),then γ~(-11)_(ks)(G)≤2k-n.This paper prove that γ~(-11)_(5s)(Q_3)=4 for the 3-Cube Q_3,and hence disprove the conjecture.In addition,we give an upper bound of the signed k-subdomination numbers for 3-regular bipartite graphs,that is,k(n2+1≤k≤n) holds for all 3-regular bipartite graphs G and positive integers γ~(-11)_(ks)(G)≤2(k+1)-n.