On the path partition of graphs

TL;DR

This paper proves a conjecture relating the path partition number of a graph to its degree parameters, confirming a specific upper bound when the maximum degree is at least twice the minimum degree.

Contribution

The paper confirms Magnant, Wang, and Yuan's conjecture on the path partition number for graphs with maximum degree at least twice the minimum degree.

Findings

Confirmed the conjecture for graphs with Δ ≥ 2δ.

Established an upper bound on the path partition number.

Provided theoretical proof for the conjecture.

Abstract

Let $G$ be a graph of order $n$. The maximum and minimum degree of $G$ are denoted by $\Delta$ and $\delta$ respectively. The \emph{path partition number} $\mu (G)$ of a graph $G$ is the minimum number of paths needed to partition the vertices of $G$. Magnant, Wang and Yuan conjectured that $\mu (G)\leq \max \left \{ \frac{n}{\delta +1}, \frac{\left( \Delta -\delta \right) n}{\left( \Delta +\delta \right) }\right \} .$ In this work, we give a positive answer to this conjecture, for $ \Delta \geq 2 \delta $.\medskip \end{abstract}