THE CLIQUE MINOR OF GRAPHS WITH INDEPENDENCE NUMBER TWO

A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. Hadwiger conjectures that every k-chromatic graph contains K-k as a minor. It implies that every graph G on n vertices has a K inverted left perpendicular (n/alpha(G))inverted right perpendicular as a minor, where alpha(G) is the independence number of G. In this paper, we consider the graphs G with independence number two. It is proved that G has a K-l-minor using prevertices of size one or two, where l = min{inverted left perpendicular n/2 inverted left perpendicular, n - k} and kappa is the connectivity of G. It is also proved that G has a K inverted left perpendicular vertical bar G vertical bar/2 right perpendicular - minar if delta (G) <= left perpendicular vertical bar G vertical bar/2 right perpendicular +1.