[Paper Review] Degree Monotone Paths

Yair Caro, Josef Lauri+1 more|arXiv (Cornell University)|May 8, 2014

Limits and Structures in Graph Theory14 references3 citations

TL;DR

This paper introduces and analyzes degree monotone paths in graphs—paths where vertex degrees are non-decreasing or non-increasing along the path. Using the Gallai-Roy theorem and connections to Turán numbers, it establishes bounds on the maximum length of such paths, derives Nordhaus-Gaddum-type inequalities for $mp(G) + mp(\overline{G})$, and proves that $mp(G) \geq \chi(G)$, with improvements for maximal outerplanar graphs. The key contribution is a sharp lower bound of $2\sqrt{n}$ for $mp(G) + mp(\overline{G})$, and a construction showing this bound is tight for certain graph families.

ABSTRACT

We shall study degree-monotone paths in graphs, a problem inspired by the celebrated theorem of Erd{ő}s-Szekeres concerning the longest monotone subsequence of a given sequence of numbers. A path P in a graph G is said to be a degree monotone path if the sequence of degrees of the vertices in P in the order they appear in P is monotonic. In this paper we shall consider these three problem related to this parameter: 1. Find bounds on $mp(G)$ in terms of other parameters of $G$. 2. Study $f(n,k)$ defined to be the maximum number of edges in a graph on $n$ vertices with $mp(G) < k$. 3. Estimate the minimum and the maximum over all graph $G$ on $n$ vertices of $mp(G)+mp(\overline{G})$. For the first problem our main tool will be the Gallai-Roy Theorem on directed paths and chromatic number. We shall also consider in some detail maximal planar and maximal outerplanar graphs in order to investigate the sharpness of the bounds obtained. For the second problem we establish a close link between $f(n,k)$ and the classical Turan numbers. For the third problem we establish some Nordhaus-Gaddum type of inequalities. We conclude by indicating some open problems which our results point to.

Motivation & Objective

• To investigate the maximum length of degree monotone paths in graphs, defined as paths where vertex degrees are non-decreasing or non-increasing along the path.
• To establish bounds on $mp(G)$, the length of the longest degree monotone path, in terms of chromatic number $\chi(G)$ and other graph parameters.
• To study $f(n,k)$, the maximum number of edges in an $n$-vertex graph with no degree monotone path of length $k$, and relate it to Turán numbers.
• To determine the minimum and maximum values of $mp(G) + mp(\overline{G})$ over all $n$-vertex graphs, extending Nordhaus-Gaddum-type results to the $mp$ parameter.

Proposed method

• Applies the Gallai-Roy theorem on directed paths and chromatic number to derive a lower bound $mp(G) \geq \chi(G)$ via a degree-based orientation of edges.
• Uses light-edge techniques in maximal outerplanar graphs to show $mp(G) \geq 4$ for $n \geq 5$, improving the general $\chi(G)$ bound.
• Establishes a connection between $f(n,k)$ and Turán numbers by constructing extremal graphs based on Albertson's earlier work.
• Applies Nordhaus-Gaddum bounds for chromatic number to derive bounds on $mp(G) + mp(\overline{G})$, proving $2\sqrt{n} \leq mp(G) + mp(\overline{G}) \leq 2n$.
• Constructs specific graph families (e.g., disjoint cliques and their complements) to demonstrate tightness of bounds.
• Uses extremal graph constructions to show that $f(n,k) \geq g(n,k)$, with a conjecture that equality holds for $n \geq \frac{(k-1)(k+2)}{2}$.

Experimental results

Research questions

• RQ1 What is the best possible lower bound for $mp(G)$ in terms of $\chi(G)$, and can it be improved for specific graph classes like maximal outerplanar graphs?
• RQ2 How does $f(n,k)$, the maximum number of edges in an $n$-vertex graph with no degree monotone path of length $k$, relate to classical Turán numbers?
• RQ3 What are the tightest possible bounds for $mp(G) + mp(\overline{G})$ over all $n$-vertex graphs, and is the lower bound $2\sqrt{n}$ sharp?
• RQ4 Can the bound $mp(G) \geq \chi(G)$ be improved for maximal planar graphs, and do there exist arbitrarily large maximal planar graphs with $mp(G) = \chi(G) = 3$?
• RQ5 Is $f(n,k)$ exactly equal to $g(n,k)$, the maximum number of edges in a graph with distinct degree sequences summing to $n$?

Key findings

• For every graph $G$, $mp(G) \geq \chi(G)$, and this bound is sharp, as shown by a complete $k$-partite graph with parts of distinct sizes.
• In maximal outerplanar graphs with $n \geq 5$ vertices, $mp(G) \geq 4$, which improves upon the $\chi(G)$-based bound and is best possible.
• There exist arbitrarily large maximal planar graphs with $\chi(G) = mp(G) = 4$, showing the $\chi(G)$ bound is tight for this class.
• The sum $mp(G) + mp(\overline{G})$ satisfies $2\sqrt{n} \leq mp(G) + mp(\overline{G}) \leq 2n$, and both bounds are sharp: the upper bound is achieved by $K_n$ minus a Hamiltonian cycle, and the lower bound is approached by a construction of disjoint cliques.
• The function $f(n,k)$, the maximum number of edges in an $n$-vertex graph with no degree monotone path of length $k$, satisfies $f(n,k) \geq g(n,k)$, and the authors conjecture equality holds for $n \geq \frac{(k-1)(k+2)}{2}$.
• A construction shows $mp(G) + mp(\overline{G}) = \frac{5\sqrt{n}}{2}$ for even square $n$, demonstrating that the $2\sqrt{n}$ lower bound is not always tight, but the asymptotic order is correct.