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On Graphs Embeddable in a Layer of a Hypercube and Their Extremal Numbers.

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This summary is machine-generated.

This study investigates cubical graphs and their Turán density in hypercubes. We characterize layered graphs and show most subdivisions have zero Turán density, but some non-layered graphs have positive density.

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Area of Science:

  • Graph theory
  • Combinatorics
  • Hypercube structures

Background:

  • Cubical graphs are subgraphs of hypercubes.
  • Turán density determines if a graph H has a positive proportion of edges in a hypercube subgraph.
  • Layered graphs are a specific subset of cubical graphs.

Purpose of the Study:

  • Characterize layered graphs within hypercubes.
  • Investigate the Turán density of various cubical graphs, particularly layered and non-layered ones.
  • Extend understanding of Turán density for cycles and subdivisions in hypercubes.

Main Methods:

  • Focusing on layered graphs within hypercubes.
  • Utilizing edge-colorings for graph characterization.
  • Analyzing subdivisions and cycles for Turán density properties.

Main Results:

  • Layered graphs are characterized by edge-colorings.
  • Most non-trivial subdivisions exhibit zero Turán density in hypercubes.
  • Discovered non-layered cubical graphs with positive Turán density and girth 8.

Conclusions:

  • Layered graphs have distinct properties within hypercubes.
  • The Turán density of the 10-cycle remains an open question, but its extremal number behaves uniquely.
  • This research advances the understanding of graph properties and density within hypercube structures.