[Paper Review] On the Curling Number of Certain Graphs

Johan Kok, Sudev Naduvath+1 more|arXiv (Cornell University)|Jun 2, 2015

Advanced Graph Theory Research7 references3 citations

TL;DR

This paper introduces the curling number and compound curling number of graphs by analyzing maximal subsequences in degree sequences, proving the curling number conjecture holds for connected simple graphs, and deriving exact formulas for set-graphs and Jaco graphs. Key results include the curling number of a set-graph being $\binom{n}{\lfloor n/2 \rfloor}$ and its compound curling number being $\prod_{i=1}^n \binom{n}{i}$, which is a perfect square iff $n$ is odd.

ABSTRACT

In this paper, we introduce the concept of curling subsequence of simple, finite and connected graphs. A curling subsequence is a maximal subsequence $C$ of the degree sequence of a simple connected graph $G$ for which the curling number $cn(G)$ corresponds to the curling number of the degree sequence per se and hence we call it the curling number of the graph $G$. A maximal degree subsequence with equal entries is called an identity subsequence. The number of identity curling subsequences in a simple connected graph $G$ is denoted $ic(G).$ We show that the curling number conjecture holds for the degree sequence of a simple connected graph $G$ on $n \geq 1$ vertices. We also introduce the notion of the compound curling number of a simple connected graph $G$ and then initiate a study on the curling number of certain standard graphs like Jaco graphs and set-graphs.

Motivation & Objective

• To extend the concept of curling number from integer sequences to degree sequences of simple, finite, connected graphs.
• To define and analyze new graph parameters: curling subsequence, curling number, identity curling subsequence, curling index, and compound curling number.
• To investigate the validity of the curling number conjecture within the context of graph degree sequences.
• To determine the curling number and compound curling number for standard graph families, particularly set-graphs and Jaco graphs.
• To explore number-theoretic properties of these new graph parameters, especially for set-graphs.

Proposed method

• Define a curling subsequence as a maximal subsequence of the degree sequence with maximum curling number.
• Introduce the curling number of a graph as the curling number of its curling subsequence.
• Define the compound curling number as the product of the multiplicities of each distinct degree in the degree sequence.
• Use combinatorial analysis of degree sequences, particularly exploiting symmetry in set-graphs where vertices of equal subset cardinality have equal degrees.
• Apply known results on binomial coefficients and hyperfactorials to derive closed-form expressions for the compound curling number.
• Leverage the identity $\binom{n}{r} = \binom{n}{n-r}$ to analyze the squareness of the compound curling number.

Experimental results

Research questions

• RQ1 Does the curling number conjecture hold for the degree sequence of any finite, simple, connected graph?
• RQ2 What is the curling number of a set-graph $G_{A^{(n)}}$ with respect to a finite set $A^{(n)}$ of size $n$?
• RQ3 What is the compound curling number of a set-graph, and when is it a perfect square?
• RQ4 Can non-regular graphs exist with equal curling and compound curling numbers, and if so, how can they be characterized?
• RQ5 How do the curling number and compound curling number of a spanning subgraph compare to those of the original regular graph?

Key findings

• The curling number conjecture holds for the degree sequence of any simple, finite, connected graph $G$ on $n \geq 1$ vertices.
• The curling number of a set-graph $G_{A^{(n)}}$ is $\binom{n}{\lfloor n/2 \rfloor}$, the largest binomial coefficient for $n$.
• The compound curling number of a set-graph $G_{A^{(n)}}$ is $\prod_{i=1}^n \binom{n}{i}$, which equals $\frac{1}{(n!)^{n+1}} \left( \prod_{i=1}^n i^i \right)^2 - 1$.
• The compound curling number of a set-graph is a perfect square if and only if $n$ is an odd integer.
• The number of curling subsequences in a graph $G$ is $1$ if $ic(G) = 1$, and $ic(G) + ic(G)!$ otherwise.
• For the degree sequence $(3,5,3,3,5,5,6)$, the curling number is $3$, achieved by subsequences like $(3,3,3)$, $(5,5,5)$, $(3,5,3,5,3,5)$, and $(5,3,5,3,5,3)$.