[Paper Review] On Equitable Coloring Parameters of Certain Cycle Related Graphs

K. P. Chithra, E. A. Shiny+1 more|arXiv (Cornell University)|Jan 1, 2018

Graph theory and applications9 references3 citations

TL;DR

This paper introduces and computes equitable coloring mean and variance—statistical parameters derived from equitable vertex colorings—for several cycle-related graphs, including wheel, flower, sunflower, closed sunflower, and blossom graphs. It provides closed-form expressions for these parameters, demonstrating their dependence on graph structure and parity of the cycle size.

ABSTRACT

Coloring the vertices of a graph G subject to given conditions can be considered as a random experiment and corresponding to this experiment, a discrete random variable X can be defined as the colour of a vertex chosen at random, with respect to the given type of colouring of G and a probability mass function for this random variable can be defined accordingly. A proper coloring C of a graph G, which assigns colors to the vertices of G such that the numbers of vertices in any two colour classes differ by at most one, is called an equitable colouring of G. In this paper, we study two statistical parameters of certain cycle related graphs, with respect to their equitable colorings.

Motivation & Objective

• To extend statistical parameters—mean and variance—into the domain of equitable graph coloring.
• To analyze equitable coloring parameters for specific cycle-related graph families, including wheel, flower, sunflower, closed sunflower, and blossom graphs.
• To establish closed-form expressions for equitable coloring mean and variance in terms of graph order and structural symmetry.
• To explore the applicability of these parameters in practical domains such as resource allocation and network optimization.
• To lay the foundation for future research on equitable coloring parameters in other graph classes and coloring types.

Proposed method

• Defines a random variable X representing the color index of a randomly selected vertex under an equitable coloring.
• Derives the probability mass function (p.m.f.) f(i) = θ(c_i)/|V(G)|, where θ(c_i) is the size of color class i.
• Computes the equitable coloring mean μ_χₑ(G) as the expected value of X: μ_χₑ(G) = Σ i·f(i).
• Computes the equitable coloring variance σ²_χₑ(G) as the variance of X: σ²_χₑ(G) = Σ i²·f(i) − (Σ i·f(i))².
• Applies these definitions to specific cycle-related graphs using structural symmetry and equitable coloring constraints.
• Derives closed-form expressions for μ_χₑ and σ²_χₑ(G) by analyzing color class distributions in graphs like W_n, F_n, SF_n, CSF_n, and Bl_n.

Experimental results

Research questions

• RQ1 What are the equitable coloring mean and variance for wheel graphs W_n, and how do they vary with the parity of n?
• RQ2 How do the equitable coloring parameters behave in flower graphs F_n, and what structural features influence their values?
• RQ3 What are the equitable coloring mean and variance for sunflower graphs SF_n and their variants (CSF_n and Bl_n), and how do they depend on n?
• RQ4 Can closed-form expressions be derived for equitable coloring parameters in cycle-related graphs with symmetric color class distributions?
• RQ5 How do the statistical parameters of equitable coloring relate to graph parameters like order and maximum degree?

Key findings

• For wheel graphs W_n, the equitable coloring mean is μ_χₑ(W_n) = (n+2)²/(4(n+1)) when n is even, and (n²+4n+7)/(4(n+1)) when n is odd.
• The equitable coloring variance of W_n is σ²_χₑ(W_n) = n(n+2)(n²+2n+4)/(48(n+1)²) for even n, and (n⁴+4n³+26n²−44n−27)/(48(n+1)²) for odd n.
• For flower graphs F_n, the equitable coloring mean is μ_χₑ(F_n) = (n+1)²/(2n+1), and the variance is σ²_χₑ(F_n) = (n⁴+2n³+2n²+n)/(3(2n+1)²).
• For sunflower graphs SF_n and closed sunflower graphs CSF_n, the equitable coloring mean is μ_χₑ = (5n+1)/(2n+1), with variance depending on the parity of n: (5n²+7n)/(2n+1)² for even n, and (5n²+3n−2)/(2n+1)² for odd n.
• For blossom graphs Bl_n, the equitable coloring mean is μ_χₑ(Bl_n) = (n+1)²/(2n+1), and the variance is σ²_χₑ(Bl_n) = (n⁴+2n³+2n²+n)/(3(2n+1)²).
• The derived formulas show that equitable coloring parameters are sensitive to graph symmetry and the parity of the cycle size, with distinct expressions for even and odd cases.