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

TL;DR

This paper derives exact formulas for chromatic feature vectors of 2-trees under bichromatic triangle constraints, enabling efficient structural analysis for network applications like fault tolerance and cryptography.

Contribution

It introduces closed-form enumeration formulas for bichromatic constrained colorings of 2-trees, connecting graph features to Fibonacci, Stirling, and Bell numbers, with practical computational methods.

Findings

Exact formulas for theta and fan graphs' chromatic features.

Efficient O(n) and O(n^2) algorithms for computing features.

Features distinguish 2-trees beyond classical chromatic polynomials.

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.