Southwest Airlines (LUV) 10-Q 2003 Q3 Quarterly report

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 Quarterly Report on Form 10-Q of Southwest Airlines Co. (the "Company") for the period ended September 30, 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: October 21, 2003

By /s/ James F. Parker

James F. Parker

Chief Executive Officer