Gallai's Conjecture for Complete and "Nearly Complete" Graphs

TL;DR

This paper investigates Gallai's Conjecture for complete and nearly complete graphs, providing explicit constructions and methods to ensure the conjecture holds after certain edge removals.

Contribution

It offers explicit path decomposition constructions for complete graphs and extends these to graphs with specific edge removals, advancing understanding of Gallai's Conjecture.

Findings

Explicit path decomposition for complete graphs satisfying Gallai's Conjecture

Proof that removing stars and certain tadpoles preserves the conjecture

Introduction of a potential general approach via non-isomorphic decompositions

Abstract

The famous Gallai's Conjecture states that any connected graph with n vertices has a path decomposition containing at most (n+1)/2 paths. In this note, we explore graphs generated from removing edges from complete graphs. We first provide an explicit construction for a path decomposition of complete graphs that satisfies Gallai's Conjecture. We then use that construction to prove that we can remove stars and certain tadpoles such that the resulting graph still satisfies Gallai's Conjecture. We also introduce a potential general approach through analyzing non-isomorphic path decompositions of complete graphs.