[Paper Review] Curling Numbers of Certain Graph Powers

C Susanth, Sunny Joseph Kalayathankal+3 more|arXiv (Cornell University)|Sep 1, 2015

Graph Labeling and Dimension Problems11 references3 citations

TL;DR

This paper investigates the curling number and compound curling number of graph powers, particularly for path graphs, complete binary trees, and n-ary trees. It derives exact formulas for these invariants under various powers, showing that the curling number of the r-th power of a path P_n depends on r relative to n/2, while for complete binary trees, the curling number is 2^h and the compound curling number is 2^(h+1 choose 2).

ABSTRACT

Given a finite nonempty sequence $S$ of integers, write it as $XY^k$, where $Y^k$ is a power of greatest exponent that is a suffix of $S$: this $k$ is the curling number of $S$. The concept of curling number of sequences has already been extended to the degree sequences of graphs to define the curling number of a graph. In this paper we study the curling number of graph powers, graph products and certain other graph operations.

Motivation & Objective

• To extend the concept of curling number from integer sequences and degree sequences to graph powers.
• To determine the curling number and compound curling number for integral powers of standard graph classes such as paths, complete binary trees, and caterpillars.
• To establish general formulas for these invariants under different graph operations and powers.
• To identify open problems for future research in graph theory involving curling number parameters.

Proposed method

• Defines the r-th power of a graph G as the graph G^r with the same vertex set, where vertices are adjacent if their distance in G is at most r.
• Analyzes the degree sequence of G^r to identify repeated entries (identity subsequences), which are used to compute the curling number.
• Applies the definition of curling number to the degree sequence of G^r, treating it as a string of integers, and computes the maximum exponent of a suffix power.
• Uses the compound curling number formula cn^c(G) = ∏k_i, where k_i are the exponents of identity subsequences in the degree sequence.
• Applies combinatorial and structural analysis to paths and trees, leveraging symmetry and level-wise degree uniformity in complete binary and n-ary trees.
• Derives closed-form expressions by classifying cases based on the relationship between r and n/2 for path graphs, and by level-wise degree counting for trees.

Experimental results

Research questions

• RQ1 What is the curling number of the r-th power of a path graph P_n for different values of r relative to n/2?
• RQ2 How do the curling number and compound curling number behave for integral powers of complete binary trees?
• RQ3 Can a general formula be derived for the curling number of the r-th power of a complete n-ary tree?
• RQ4 What are the curling number and compound curling number of an arbitrary caterpillar graph and its powers?
• RQ5 What are the general properties of curling number and compound curling number under graph operations such as powers and products?

Key findings

• For the r-th power of a path P_n with r ≤ ⌊n/2⌋, the curling number is 2 if r = ⌊n/2⌋, n−2r if r < ⌊n/2⌋−1, and 2(r+1)−n if ⌊n/2⌋ ≤ r ≤ n−1.
• The curling number of the r-th power of a complete binary tree of height h is 2^h, and its compound curling number is 2^(h+1 choose 2).
• For a complete n-ary tree of height h, the curling number of its r-th power is n^h, and the compound curling number is ∏_{i=0}^h n^i.
• The curling number of a caterpillar graph G is max{η, ∑l_i}, where η is the maximum frequency of a degree among internal vertices and ∑l_i is the total number of leaves.
• The compound curling number of an arbitrary caterpillar is difficult to compute due to insufficient structural constraints on internal vertex degrees.
• The curling number of G^r for any graph G with diameter d is |V(G)| when r ≥ d, since G^r becomes a complete graph.