Isoperimetry, stability, and irredundance in direct products

Noga Alon, Colin Defant

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

The direct product of graphs G1,…,Gn is the graph with vertex set V(G1)×⋯×V(Gn) in which two vertices (g1,…,gn) and (g1 ,…,gn ) are adjacent if and only if gi is adjacent to gi in Gi for all i. Building off of the recent work of Brakensiek, we prove an optimal vertex isoperimetric inequality for direct products of complete multipartite graphs. Applying this inequality, we derive a stability result for independent sets in direct products of balanced complete multipartite graphs, showing that every large independent set must be close to the maximal independent set determined by setting one of the coordinates to be constant. Armed with these isoperimetry and stability results, we prove that the upper irredundance number of a direct product of balanced complete multipartite graphs is equal to its independence number in all but at most 37 cases. This proves most of a conjecture of Burcroff that arose as a strengthening of a conjecture of the second author and Iyer. We also propose a further strengthening of Burcroff's conjecture.

Original languageEnglish (US)
Article number111902
JournalDiscrete Mathematics
Volume343
Issue number7
DOIs
StatePublished - Jul 2020

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Keywords

  • Complete multipartite graph
  • Direct product graph
  • Domination
  • Independent set
  • Irredundance
  • Isoperimetry

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