[Paper Review] On Chromatic Core Subgraph of Simple Graphs

Johan Kok, Sudev Naduvath+2 more|arXiv (Cornell University)|Mar 5, 2018

Optimization and Packing Problems4 references3 citations

TL;DR

This paper introduces the chromatic core subgraph—a minimal induced subgraph preserving the chromatic number of a simple graph—by minimizing the structor index (ν + ε). It establishes existence conditions, characterizes chromatic cores for classical graphs like cycles, wheels, and trees, and proves that line graphs of trees have maximum cliques as chromatic cores, offering a framework for resilient network design under destruction while maintaining full technological diversity.

ABSTRACT

If distinct colours represent distinct technology types that are placed at the vertices of a simple graph in accordance to a minimum proper colouring, a disaster recovery strategy could rely on an answer to the question: "What is the maximum destruction, if any, the graph (a network) can undergo while ensuring that at least one of each technology type remain, in accordance to a minimum proper colouring of the remaining induced subgraph." In this paper, we introduce the notion of a chromatic core subgraph $H$ of a given simple graph $G$ in answer to the stated problem. Since for any subgraph $H$ of $G$ it holds that $χ(H) \leq χ(G)$, the problem is well defined.

Motivation & Objective

• To define and formalize the chromatic core subgraph as the smallest induced subgraph preserving the chromatic number of a graph.
• To address disaster recovery in network design by ensuring at least one vertex of each color class remains after partial destruction.
• To characterize chromatic core subgraphs for classical graph families such as paths, cycles, wheels, and trees.
• To extend the concept to line graphs of trees and explore structural properties via the structor index (ν + ε).
• To lay foundations for generalizing the notion to edge colorings and other derived coloring schemes.

Proposed method

• Define the chromatic core subgraph H of a graph G as the induced subgraph with minimum structor index (ν(H) + ε(H)) such that χ(H) = χ(G).
• Use Brook’s Theorem to establish that only complete graphs and odd cycles lack proper subgraphs with the same chromatic number.
• Apply component-wise analysis: for a graph with multiple components, the chromatic core is formed by the chromatic cores of the highest-chromatic components.
• Characterize chromatic cores for specific graph families using structural analysis and degree sequence properties.
• Prove that for trees, the line graph L(T) has a maximum clique as its chromatic core, based on the maximum degree of T.
• Use the rainbow neighborhood convention and weak perfection to derive results on independent sets and their complements in the context of chromatic cores.

Experimental results

Research questions

• RQ1 Which induced subgraphs of a given graph G preserve χ(G) while minimizing the structor index ν(H) + ε(H)?
• RQ2 Under what conditions does a graph G admit a proper subgraph H with χ(H) = χ(G)?
• RQ3 What is the chromatic core subgraph of the line graph of a tree, and how does it relate to the maximum degree of the original tree?
• RQ4 How do chromatic core subgraphs behave in disconnected graphs, particularly when components differ in chromatic number?
• RQ5 Can the concept of chromatic core subgraphs be generalized to non-simple graphs and product graphs like strong or lexicographic products?

Key findings

• The chromatic core subgraph exists for every finite, undirected, connected simple graph, and is unique up to isomorphism only under specific conditions.
• For any graph G with χ(G) = 1, the chromatic core is the single-vertex graph K₁.
• For bipartite graphs (χ(G) = 2), any edge uv is a chromatic core subgraph, so the chromatic core has ν = 2 and ε = 1.
• Odd cycles Cₙ (n ≥ 3) and complete graphs Kₙ have themselves as their unique chromatic core subgraphs.
• For even wheels Wₙ (n ≥ 4), the chromatic core is C₃; for odd wheels (n ≥ 3), the chromatic core is Wₙ itself.
• The line graph of a tree T has a maximum clique as its chromatic core, where the clique size equals the maximum degree of T.