Improving an upper bound on the size of k-regular induced subgraphs

This paper deals with the NP-hard graph problem that consists on determining the maximum cardinality subset of vertices inducing a k-regular subgraph. For any graph G, this maximum will be denoted by αk(G). From a well known Motzkin-Straus result, it is deduced a relationship between αk(G) and the independence number α(G). Next, it is proved that the upper bounds υk(G) introduced in Cardoso, Kamiński, and Lozin [J. Comb. Optim., 14, 455463, 2007] can easily be computed from υ0(G), for any positive integer k. This relationship also allows to present an alternative proof of the Hoffman bound extension introduced in the above paper. The paper continues with the introduction of a new upper bound on αk(G) improving υk(G). Due to the hardness of computing this improved bound, two methods are provided for approximating it. Finally, some computational experiments performed to compare all studied bounds are reported.