The Saturation Number for the length of Degree Monotone Paths

TL;DR

This paper investigates the saturation number related to the length of degree monotone paths in graphs, providing bounds, exact values for specific cases, and constructions of saturated graphs.

Contribution

It introduces and analyzes the saturation problem for the degree monotone path parameter, offering bounds, exact solutions for certain cases, and new graph constructions.

Findings

Linear bounds for the saturation number h(n,k).

Exact values of h(n,k) for k=3 and 4.

Constructed examples of saturated graphs.

Abstract

A degree monotone path in a graph $G$ is a path $P$ such that the sequence of degrees of the vertices in the order in which they appear on $P$ is monotonic. The length of the longest degree monotone path in $G$ is denoted by $mp(G)$. This parameter, inspired by the well-known Erdos-Szekeres theorem, has been studied by the authors in two earlier papers. Here we consider a saturation problem for the parameter $mp(G)$. We call $G$ saturated if, for every edge $e$ added to $G$, $mp(G+e) >mp(G)$, and we define $h(n,k)$ to be the least possible number of edges in a saturated graph $G$ on $n$ vertices with $mp(G) < k$, while $mp(G+e) \geq k$ for every new edge $e$. We obtain linear lower and upper bounds for $h(n,k)$, we determine exactly the values of $h(n,k)$ for $k=3$ and $4$, and we present constructions of saturated graphs.