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February 26, 2026Annals of CombinatoricsOpen Access

Counting Subgraphs of Coloring Graphs Using Shadow Graphs

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Authors

SMSimon MacLean

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Overview

This analysis shows polynomial behavior of induced copies in k-coloring graphs, suggesting new insights on graph properties.

Key Points

  • The research aims to explicitly construct H-polynomials for subgraphs within k-coloring graphs using shadow graphs.
  • Constructed k-coloring graphs from proper colorings of a given graph G.
  • Utilized shadow graphs to express H-polynomials in terms of chromatic polynomials.
  • Computed explicit formulas for H-polynomials when H is a hypercube.
  • Proved that the number of induced copies of subgraphs in k-coloring graphs is a polynomial function of k.
  • Demonstrated explicit formulas for H-polynomials for trees and hypercubes.
  • Disproved a conjecture regarding chromatic pairs polynomials using examples of two different graphs.

Cite This Study

Simon MacLean (2026) studied this question.

synapsesocial.com/papers/699fe41d95ddcd3a253e85abhttps://doi.org/10.1007/s00026-026-00809-x
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