Approximating the maximum size of a k-regular induced subgraph by an upper bound on the co-k-plex number

Abstract

Let α _k and ${\hat{\alpha }_k}$ denote respectively the maximum cardinality of a k-regular induced subgraph and the co-k-plex number of a given graph. In this paper, we introduce a convex quadratic programming upper bound on ${\hat{\alpha }_k}$, which is also an upper bound on α _k . The new bound denoted by ${\hat{\upsilon }_k}$ improves the bound υ _k given in [3]. For regular graphs, we prove a necessary and sufficient condition under which ${\hat{\upsilon }_k}$ equals υ _k . We also show that the graphs for which ${\hat{\alpha }_k}$ equals ${\hat{\upsilon }_k}$ coincide with those such that α _k equals υ _k . Next, an improvement of ${\hat{\upsilon }_k}$ denoted by ${\hat{\vartheta }_k}$ is proposed, which is not worse than the upper bound ϑ _k for α _k introduced in [8]. Finally, some computational experiments performed to appraise the gains brought by ${\hat{\vartheta }_k}$ are reported.