If you've read the U.S. Constitution clearly - or at all - you'll notice that the national government does not run elections - it gives that responsibility to the states. The U.S. Constitution does establish elected positions, which states are responsible for filling.
Here are the relevant parts of the U.S. Constitution. Notice the repeated mention of states.
Article One:
Section Two:
1. The House of Representatives shall be composed of members chosen every second year by the people of the several States, and the elector in each State shall have the qualifications requisite for electors of the most numerous branch of the State Legislature.
3. Representatives and direct taxes shall be apportioned among the several States which may be included within this Union, according to their respective numbers . . . The actual enumeration shall be made within three years after the first meeting of the Congress of the United States, and within every subsequent term of ten years, in such manner as they shall by law direct.
4. When vacancies happen in the representation from any State, the Executive Authority thereof shall issue writs of election to fill such vacancies.
Section Three:
1. The Senate of the United States shall be composed of two Senators from each State, (chosen by the Legislature thereof,) (The preceding five words were superseded by Amendment XVII) for six years; and each Senator shall have one vote.
2. Immediately after they shall be assembled in consequence of the first election, they shall be divided as equally as may be into three classes. The seats of the Senators of the first class shall be vacated at the expiration of the second year, of the second class at the expiration of the fourth year, and of the third class at the expiration of the sixth year, so that one-third may be chosen every second year; and if vacancies happen by resignation, or otherwise, during the recess of the Legislature of any State, the Executive thereof may make temporary appointments until the next meeting of the Legislature, which shall then fill such vacancies.
Section Four:
1. The times, places and manner of holding elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by law make or alter such regulations, except as to the places of choosing Senators.
Article Two:
2. Each State shall appoint, in such manner as the Legislature may direct, a number of electors, equal to the whole number of Senators and Representatives to which the State may be entitled in the Congress: but no Senator or Representative, or person holding an office of trust or profit under the United States, shall be appointed an elector.
- See also the 12th Amendment.
In case you'd like to know where, in Texas, those laws exist, you can look here:
- Texas Election Code.
