On Path Decomposition Conjecture of Tibor Gallai

TL;DR

This paper proves Gallai's Path Decomposition Conjecture for all connected graphs by first establishing it for trees, then extending the result using induction and spanning trees, and finally exploring Hamiltonian path covers in complete graphs.

Contribution

It provides a proof of Gallai's Path Decomposition Conjecture for all connected graphs, including a new bound and an alternative proof approach using spanning trees.

Findings

Gallai's P.D.C. is valid for trees.

A new bound for the number of paths in a path cover.

Existence of Hamiltonian path cover satisfying Gallai bound in complete graphs.

Abstract

We settle the Path Decomposition Conjecture (P.D.C.) due to Tibor Gallai for minimally connected graphs, i.e. trees. We use this validity for trees and settle the P. D. C. using induction on the number of edges for all connected graphs. We then obtain a new bound for the number of paths in a path cover in terms of the number of edges using idea of associating a tree with a connected graph. We then make use of a spanning tree in the given connected graph and its associated basic path cover to settle the conjecture of Tibor Gallai in an alternative way. Finally, we show the existence of Hamiltonian path cover satisfying Gallai bound for complete graphs of even order and discuss some of its possible ramifications.