Exhibit 32.1

CERTIFICATION PURSUANT TO 18 U.S.C. SECTION 1350,
AS ADOPTED PURSUANT TO
SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

     In connection with the Annual Report on Form 10-K of Southwest Airlines Co. (the “Company”) for the period ended December 31, 2003 as filed with the Securities and Exchange Commission on the date hereof (the “Report”), I, James F. Parker, Chief Executive Officer of the Company, certify pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

     (1)     The Report fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934, as amended; and

     (2)     The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of the Company.

Date: January 29, 2004

By	/s/ James F. Parker
	James F. Parker Chief Executive Officer