The Turan number of the balanced double star S_{n-1,n-1} in the hypercube Q_n

TL;DR

This paper determines the exact maximum number of edges in a hypercube graph that avoids containing a specific balanced double star subgraph, filling a gap in Turan number research for hypercubes.

Contribution

It provides the first exact calculation of the Turan number for the balanced double star in hypercubes, a previously unexplored problem.

Findings

Exact Turan number for S_{n-1,n-1} in Q_n is 2^{n-3}*(4n-3) for all n >= 3.

Establishes a new result in extremal graph theory within hypercube structures.

Extends Turan number studies from planar graphs to hypercube graphs.

Abstract

The n-dimensional hypercube Q_n is a graph with vertex set {0,1}^n such that there is an edge between two vertices if and only if they differ in exactly one coordinate. Let H be a graph, and a graph is called H-free if it does not contain H as a subgraph. Given a graph H, the Turan number of H in Q_n, denoted by ex(Q_n, H), is the maximum number of edges of a subgraph of Q_n that is H-free. A double star S_{k,l} is the graph obtained by taking an edge uv and joining u with k vertices, v with l vertices which are different from the k vertices. We say a double star is a balanced double star if k = l. Currently, the Turan number of the balanced star S_{n,n} is has been studied in the planar graphs. In the hypercubes, the researchers look for the maximum number of edges of the graphs that are C_k-free. However, the Turan number of the double star in the hypercube remains unexplored. Building upon prior research, we initiate the first study on the Turan number of the balanced double star in the hypercube. In this paper, we give the exact value of the Turan number of the balanced double star S_{n-1,n-1} in the hypercube Q_n, which is 2^{n-3}*(4n- 3) for all n >= 3.