A Kₜ-expansion consists of t vertex-disjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be two-colored so that the edges of the trees are bichromatic but the edges between trees are monochromatic. A graph contains an odd Kₜ minor or an odd clique minor of order t if it contains an odd Kₜ-expansion. Gerards and Seymour from 1995 conjectured that every graph G contains an odd K (₆) minor, where (G) denotes the chromatic number of G. This conjecture is referred to as ``Odd Hadwiger's Conjecture". Let (G) denote the independence number of a graph G. In this paper we investigate the Odd Hadwiger's Conjecture for graphs G with (G) 2. We first observe that a graph G on n vertices with (G) 2 contains an odd K (₆) minor if and only if G contains an odd clique minor of order n/2. We then prove that every graph G on n vertices with (G) 2 contains an odd clique minor of order n/2 if G contains a clique of order n/4 when n is even and (n+3) /4 when n is odd, or G does not contain H as an induced subgraph, where (H) 2 and H is an induced subgraph of K₁ + P₄, K₂+ (K₁ K₃), K₁+ (K₁ K₄), K₇^-, K₇, or the kite graph.
Ji et al. (Mon,) studied this question.