Seymour and Woodall's conjecture holds for graphs with independence number two

Woodall (and Seymour independently) in 2001 proposed a conjecture that every graph G contains every complete bipartite graph on χ(G) vertices as a minor, where χ(G) is the chromatic number of G. In this paper, we prove that for each positive integer with 2 ≤ χ(G), each graph G with independence number two contains a K^,χ(G)--minor, implying that Seymour and Woodall's conjecture holds for graphs with independence number two, where K^,χ(G)- is the graph obtained from K,χ(G)- by making every pair of vertices on the side of the bipartition of size adjacent.