Density of linearity index in the interval of matching numbers

Abstract

Given integers 2 <= p <= c <= q, we construct a finite simple graph G with nu(1)(G) = p and nu(G) = q for which the square free power I(G)([k] )of the edge ideal I(G) of G has linear quotients for each c <= k <= q and is not linearly related for each 1 <= k < c, where nu(1)(G)is the induced matching number of G and nu(G) is the matching number of G.