FOURTH SUPPLEMENTAL INDENTURE

Exhibit 4.11

FOURTH SUPPLEMENTAL INDENTURE

This Fourth Supplemental Indenture (this “Supplemental Indenture”), dated as of May 3, 2011, is entered into by and among Southwest Airlines Co., a Texas corporation (“Southwest”), Pedernales Debt Sub, LLC, a Texas limited liability company (the “Successor Company”), and U.S. Bank National Association, a national banking association, as trustee hereunder (the “Trustee”). Capitalized terms used herein and not otherwise defined have the meanings set forth in the Indenture (as defined below).

WHEREAS, AirTran Holdings, Inc., a Nevada corporation (“AirTran Holdings Inc”), and the Trustee were the original parties to a Senior Indenture dated as of October 14, 2009 (the “Base Indenture”) and a First Supplemental Indenture dated as of October 14, 2009 relating to the issuance of the 5.25% Convertible Senior Notes due 2016 (the “First Supplemental Indenture”);

WHEREAS, AirTran Holdings Inc, Southwest and the Trustee were the original parties to a Second Supplemental Indenture dated as of May 2, 2011, which supplements the Base Indenture and the First Supplemental Indenture (the “Second Supplemental Indenture”), pursuant to which Second Supplemental Indenture (among other things) Southwest fully and unconditionally guaranteed all obligations of AirTran Holdings Inc, its wholly owned subsidiary, under the Securities and the Indenture;

WHEREAS, Southwest, AirTran Holdings, LLC, a Texas limited liability company (“AirTran LLC”), and the Trustee entered into a Third Supplemental Indenture dated as of May 2, 2011, which supplements the Base Indenture, the First Supplemental Indenture, and the Second Supplemental Indenture (the “Third Supplemental Indenture” and, the Base Indenture as supplemented by the First Supplemental Indenture, the Second Supplemental Indenture, and the Third Supplemental Indenture, the “Indenture”), pursuant to which Third Supplemental Indenture (among other things) AirTran LLC, as successor by merger to AirTran Holdings Inc, assumed all of the obligations of the Company under the Indenture and Securities;

WHEREAS, AirTran LLC, the Successor Company and Pedernales Asset Sub, LLC, a Texas limited liability company (“TX Asset Sub”), entered into the Agreement and Plan of Merger dated as of May 3, 2011 (the “Merger Agreement”), pursuant to which, at the effective time of the merger contemplated thereby, AirTran LLC, the Successor Company and TX Asset Sub merged with and into each other and as a result of the multi-survivor merger, the Successor Company and AirTran LLC survived the merger with the Successor Company being the obligor under the Indenture and the Notes;

WHEREAS, pursuant to Section 6.01 of the First Supplemental Indenture, AirTran LLC, as the Company under the Indenture, shall not consolidate with, or merge into, any other Person unless the surviving Person is a corporation or limited liability company organized and validly existing under the laws of the United States or any state thereof, and such Person expressly assumes, by an indenture supplemental to the Indenture, executed and delivered to the Trustee, in form reasonably satisfactory to the Trustee, all of the obligations of the Company under the Securities and the Indenture;

1

WHEREAS, in accordance with Section 8.01 of the First Supplemental Indenture, the Successor Company, Southwest and the Trustee may amend or supplement the Indenture as provided in this Supplemental Indenture without notice to, or consent of, any Securityholder;

WHEREAS, each of Southwest and the Successor Company have duly authorized the execution and delivery of this Supplemental Indenture;

WHEREAS, the Successor Company has furnished the Trustee with an Opinion of Counsel and an Officer’s Certificate in accordance with the Indenture, stating that the execution of this Supplemental Indenture is authorized or permitted by the Indenture; and

WHEREAS, all things necessary to make this Supplemental Indenture a valid agreement of the Successor Company, Southwest and the Trustee and a valid amendment of, and supplement to, the Indenture have been done, and the entry into this Supplemental Indenture by the parties hereto is in all respects authorized by the provisions of the Indenture.

NOW, THEREFORE, in consideration of the premises set forth herein and for other good and valuable consideration, the receipt of which is hereby acknowledged, the parties hereto covenant and agree for the equal and proportionate benefit of all Holders of the Securities, as follows:

ARTICLE I

CONFIRMATION OF INDENTURE; ASSUMPTION

Section 1.01 Confirmation of Indenture. Except as supplemented hereby, the Indenture, including, without limitation, Southwest’s full and unconditional guaranty of all the obligations of the Company under the Indenture and Securities, is hereby ratified, confirmed, and reaffirmed in all respects. The Indenture and this Supplemental Indenture shall be read, taken, and construed as one and the same instrument.

Section 1.02 Assumption. Pursuant to, and in compliance and in accordance with, Section 6.01 of the First Supplemental Indenture, the Successor Company hereby expressly and unconditionally assumes all of the obligations of the Company under the Securities and the Indenture, including, without limitation, the due and punctual payment of the principal of, premium, if any, and interest on, the Securities and the due and punctual observance of each and every covenant and condition of the Company under the Indenture, all as if the Successor Company were originally the Company thereunder.

Section 1.03 Successor Substituted. In accordance with Section 6.02 of the First Supplemental Indenture, upon the merger of AirTran LLC into the Successor Company, the Successor Company succeeded to, and was substituted for, and may exercise every right and power of, AirTran LLC as the “Company” under the Indenture with the same effect as if the Successor Company had been named as the Company in the Indenture.

Section 1.04 Representations and Warranties. The Successor Company represents that (a) it has all necessary power and authority to execute and deliver this Supplemental Indenture and to perform the Indenture, (b) it is the successor by merger to AirTran LLC pursuant to a valid merger effected in accordance with applicable law, (c) it is a limited liability company

2

organized and validly existing under the laws of the State of Texas, (d) both before and immediately after giving effect to this Supplemental Indenture, no Default or Event of Default has or will have occurred or be continuing, and (e) this Supplemental Indenture is executed and delivered pursuant to Section 8.01 of the First Supplemental Indenture and does not require the consent of Securityholders.

ARTICLE II

MISCELLANEOUS PROVISIONS

Section 2.01 Concerning the Trustee. The Trustee assumes no duties, responsibilities, or liabilities by reason of this Supplemental Indenture other than as set forth in the Indenture. The Trustee shall not be responsible in any manner whatsoever for or in respect of the validity or sufficiency of this Supplemental Indenture or for or in respect of the recitals contained herein, all of which recitals are made solely by Southwest and the Successor Company. In addition, and without limiting the foregoing, the Trustee is not charged with knowledge of the Merger Agreement or any of the terms thereof.

Section 2.02 Supplemental Indenture Controls. In the event of a conflict or inconsistency between the Indenture and this Supplemental Indenture, the provisions of this Supplemental Indenture shall control.

Section 2.03 Governing Law. THIS SUPPLEMENTAL INDENTURE AND THE SECURITIES SHALL BE GOVERNED BY, AND CONSTRUED IN ACCORDANCE WITH, THE LAWS OF THE STATE OF NEW YORK.

Section 2.04 No Adverse Interpretation of Other Agreements. This Supplemental Indenture may not be used to interpret another indenture, loan, or debt agreement other than the Indenture for purposes of the Securities. Any such indenture, loan, or debt agreement may not be used to interpret this Supplemental Indenture.

Section 2.05 Multiple Counterparts. The parties may sign multiple counterparts of this Supplemental Indenture. Each signed counterpart shall be deemed an original, but all of them together shall represent the same agreement.

Section 2.06 Headings. The titles and headings of the articles and sections of this Supplemental Indenture have been inserted for convenience of reference only, are not to be considered a part hereof, and shall in no way modify or restrict any of the terms or provisions hereof.

Section 2.07 Successors and Assigns. All covenants and agreements made by the Successor Company and Southwest in this Supplemental Indenture shall be binding upon their respective successors and assigns, whether expressed or not.

[Signature Page Follows]

3

IN WITNESS WHEREOF, the parties hereto have caused this Supplemental Indenture to be duly executed.

Signature Page to Fourth Supplemental Indenture

IN WITNESS WHEREOF, the parties hereto have caused this Supplemental Indenture to be duly executed.

Signature Page to Fourth Supplemental Indenture

IN WITNESS WHEREOF, the parties hereto have caused this Supplemental Indenture to be duly executed.

Signature Page to Fourth Supplemental Indenture