Chromatic Feature Vectors for 2-Trees: Exact Formulas for Partition Enumeration with Network Applications

TL;DR

The paper derives exact closed-form expressions for chromatic feature vectors of two fundamental 2-tree families under a bichromatic triangle constraint, linking the counts to classical combinatorial sequences. For Theta_n, it yields $r_2(\Theta_n)=2^{n-2}+1$ and $r_k(\Theta_n)=S(n-2,k-1)$ for $k\ge3$, computable in linear time with precomputed Stirling numbers. For Phi_n, it establishes $r_2(\Phi_n)=F_{n+1}$ and provides a closed formula $r_k(\Phi_n)=\sum_{t=k-1}^{n-1} a_{n-1,t} S(t,k-1)$ with $a_{m,t}$ given by binomial coefficients, achieving $O(n^2)$ computation per component. The results connect to Bell numbers and Fibonacci polynomials, offering informative, polynomial-time features that complement traditional graph embeddings and have applications in distributed systems and cryptography.

Abstract

We establish closed-form enumeration formulas for chromatic feature vectors of 2-trees under the bichromatic triangle constraint. These efficiently computable structural features derive from constrained graph colorings where each triangle uses exactly two colors, forbidding monochromatic and rainbow triangles, a constraint arising in distributed systems where components avoid complete concentration or isolation. For theta graphs Theta_n, we prove r_k(Theta_n) = S(n-2, k-1) for k >= 3 (Stirling numbers of the second kind) and r_2(Theta_n) = 2^(n-2) + 1, computable in O(n) time. For fan graphs Phi_n, we establish r_2(Phi_n) = F_{n+1} (Fibonacci numbers) and derive explicit formulas r_k(Phi_n) = sum_{t=k-1}^{n-1} a_{n-1,t} * S(t, k-1) with efficiently computable binomial coefficients, achieving O(n^2) computation per component. Unlike classical chromatic polynomials, which assign identical features to all n-vertex 2-trees, bichromatic constraints provide informative structural features. While not complete graph invariants, these features capture meaningful structural properties through connections to Fibonacci polynomials, Bell numbers, and independent set enumeration. Applications include Byzantine fault tolerance in hierarchical networks, VM allocation in cloud computing, and secret-sharing protocols in distributed cryptography.

Theorems & Definitions (33)

• Definition 1: Graph Families
• Definition 2: Chromatic Feature Vector
• Definition 3: Bichromatic Triangle Coloring Polynomial
• Definition 4: Stirling Numbers of the Second Kind
• Definition 5: Fibonacci Numbers
• Proposition 1: Uniformity of Classical Partition Vectors
• proof
• Theorem 2
• proof
• Corollary 3