On even-cycle-free subgraphs of the doubled Johnson graphs

Abstract

The generalized Turán number ex(G,H) is the maximum number of edges in an H-free subgraph of a graph G. It is an important extension of the classical Turán number ex(n,H), which is the maximum number of edges in a graph with n vertices that does not contain H as a subgraph. In this paper, we consider the maximum number of edges in an even-cycle-free subgraph of the doubled Johnson graphs J(n;k,k+1), which are bipartite subgraphs of hypercube graphs. We give an upper bound for ex(J(n;k,k+1),C2r) with any fixed k∈Z+ and any n∈Z+ with n≥2k+1. We also give an upper bound for ex(J(2k+1;k,k+1),C2r) with any k∈Z+, where J(2k+1;k,k+1) is known as doubled Odd graph O˜k+1. This bound induces that the number of edges in any C2r-free subgraph of O˜k+1 is o(e(O˜k+1)) for r≥6, which also implies a Ramsey-type result.