Estimate of the Number of Edges in Subgraphs of a Johnson Graph

F A Pushnyakov,A M Raigorodskii

doi:10.1134/s106456242104013x

Abstract

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained.

Highlights

New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained
Recall that an independent vertex set of a graph G is any subset of its vertices such that no two vertices in the subset represent an edge of G, and the independence number α(G) of G is equal to the cardinality of any maximum independent vertex set of G
The classical Turán theorem states that the following estimate is sharp on the set of all graphs: a Moscow Institute of Physics and Technology (National Research University), Dolgoprudnyi, Moscow oblast, 141701 Russia b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, 119991 Russia c Caucasus Mathematical Center, Adyghe State University, Maykop, 385000 Republic of Adygea, Russia d Institute of Mathematics and Computer Science, Buryat State University, Ulan-Ude, 670000 Buryat Republic, Russia

Summary

Introduction

Abstract—New estimates for the minimum number of edges in subgraphs of a Johnson graph are obtained. E = {(A, B): A ∩ B = s}, i.e., the graph vertices are all possible r-element subsets of [n] := {1,..., n} and the edges join pairs of sets intersecting exactly in s elements. Recall that an independent vertex set of a graph G is any subset of its vertices such that no two vertices in the subset represent an edge of G, and the independence number α(G) of G is equal to the cardinality of any maximum independent vertex set of G.

Results

Conclusion