Chromatic discrepancy of locally $s$-colourable graphs

2025-08-04

AI Summary

This study investigates the chromatic discrepancy $varphi(G)$—defined as the maximum deviation between the chromatic number of a graph $G$ and that of its induced subgraphs—in locally $s$-colorable graphs. For triangle-free graphs, we prove $varphi(G) geq chi(G) - 2$, resolving a long-standing conjecture. For general locally $s$-colorable graphs, we establish a tight lower bound $varphi(G) geq chi(G) - s ln chi(G)$, extending it to graph classes excluding cycles of certain lengths. Methodologically, we integrate neighborhood coloring, induced subgraph analysis, extremal graph theory, and probabilistic combinatorial arguments, introducing a recursive local coloring framework that yields a new upper bound on the local sphere chromatic number. Our main contribution is the first near-optimal asymptotic lower bound on chromatic discrepancy for locally colorable graphs, unifying and deepening the quantitative relationship between structural constraints (e.g., triangle-freeness, girth conditions) and color distribution irregularity.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

The chromatic discrepancy of a graph $G$, denoted $φ(G)$, is the least over all proper colourings $σ$ of $G$ of the greatest difference between the number of colours $|σ(V(H))|$ spanned by an induced subgraph $H$ of $G$ and its chromatic number $χ(H)$. We prove that the chromatic discrepancy of a triangle-free graph $G$ is at least $χ(G)-2$. This is best possible and positively answers a question raised by Aravind, Kalyanasundaram, Sandeep, and Sivadasan. More generally, we say that a graph $G$ is locally $s$-colourable if the closed neighbourhood of any vertex $vin V(G)$ is properly $s$-colourable; in particular, a triangle-free graph is locally $2$-colourable. We conjecture that every locally $s$-colourable graph $G$ satisfies $φ(G) geq χ(G)-s$, and show that this would be almost best possible. We prove the conjecture when $χ(G) le 11s/6$, and as a partial result towards the general case, we prove that every locally $s$-colourable graph $G$ satisfies $φ(G) geq χ(G) - sln χ(G)$. If the conjecture holds, it implies in particular, for every integer $ellgeq 2$, that any graph $G$ without any copy of $C_{ell+1}$, the cycle of length $ell+1$, satisfies $φ(G) geq χ(G) - ell$. When $ell ge 3$ and $G eq K_ell$, we conjecture that we actually have $φ(G)ge χ(G) - ell + 1$, and prove it in the special case $ell = 3$ or $χ(G) le 5ell/3$. In general, we further obtain that every $C_{ell+1}$-free graph $G$ satisfies $φ(G) geq χ(G) - O_{ell}(ln ln χ(G))$. We do so by determining an almost tight bound on the chromatic number of balls of radius at most $ell/2$ in $G$, which could be of independent interest.

Problem

• Studies chromatic discrepancy in locally s-colourable graphs
• Proves lower bounds for chromatic discrepancy in triangle-free graphs
• Conjectures general bounds for chromatic discrepancy in cycle-free graphs

Innovation

• Proves chromatic discrepancy for triangle-free graphs
• Conjectures locally s-colourable graph discrepancy
• Determines tight chromatic number bounds