Semi-regular graphs of minimum independence number Original Research Article

Many functions of the degree sequence of a graph give lower bounds on the graphʹs independence number. In particular, α(G)⩾R(d(G)), where R is the residue of the degree sequence of G. We consider the precision of this estimate when it is applied to semi-regular degree sequences, showing that the residue nearly always gives the best possible estimate on independence number: when d is semi-regular and graphic, we construct a graph G realizing d with R(d)⩽α(G)⩽R(d)+1. Moreover, we determine explicitly which inequality is strict. We prove this directly for most semi-regular sequences, giving an outline of proof for the remainder.