Basis for a vector space generated by Hamiltonian time paths in a complete time graph

TL;DR

This paper introduces the concept of complete time graphs and Hamiltonian time paths, analyzing the vector space they generate and determining its dimension for n ≥ 5, along with an algorithm to construct a basis.

Contribution

It defines complete time graphs and Hamiltonian time paths, and determines the dimension of the generated vector space, providing an algorithm for basis construction.

Findings

Dimension of the vector space for n ≥ 5 is determined.

An algorithm with complexity for basis construction is provided.

The characteristic functions of Hamiltonian time paths form a basis.

Abstract

In this paper we introduce the notion of a complete time graph of order n. We define time paths and Hamiltonian time paths in a complete time graph. Each Hamiltonian time path (htp) is associated with some permutation p of the integers 1 to n. The characteristic function of this path forms a vector in the vector space of rational-valued functions on the set of edges of the compete time graph. We will consider the vector space generated by these functions. The main result in this paper is to determine the dimension of this vector space for n greater than or equal to 5. We also give an algorithm with its complexity for the construction of a basis in this vector space.