Induced path factors of regular graphs

TL;DR

This paper investigates the minimum number of induced paths needed to cover all vertices in regular graphs, establishing bounds and limits for various degrees, including cubic graphs and asymptotic behavior as degree increases.

Contribution

It provides new bounds on the induced path number for regular graphs and proves the existence of a limit for the maximum induced path number ratio as the number of vertices grows.

Findings

For connected cubic graphs, the induced path number is at most (n-1)/3.

The limit of the maximum induced path number ratio exists for all regular degrees.

As degree increases, the ratio approaches 1/2, with specific bounds for degrees 3 and 4.

Abstract

An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $\rho(G)$ of $G$ is the minimum number of paths in an induced path factor of $G$. We show that if $G$ is a connected cubic graph on $n>6$ vertices, then $\rho(G)\le(n-1)/3$. Fix an integer $k\ge3$. For each $n$, define $\mathcal{M}_n$ to be the maximum value of $\rho(G)$ over all connected $k$-regular graphs $G$ on $n$ vertices. As $n\rightarrow\infty$ with $nk$ even, we show that $c_k=\lim(\mathcal{M}_n/n)$ exists. We prove that $5/18\le c_3\le1/3$ and $3/7\le c_4\le1/2$ and that $c_k=\frac12-O(k^{-1})$ for $k\rightarrow\infty$.