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

TL;DR

This paper establishes upper bounds on the maximum edges in even-cycle-free subgraphs of doubled Johnson graphs, extending Turán-type results and implying Ramsey-type conclusions for these bipartite hypercube subgraphs.

Contribution

It provides new upper bounds for the generalized Turán number in doubled Johnson graphs and their special case, the doubled Odd graph, for even cycles.

Findings

Upper bounds for ex(J(n;k,k+1), C_{2r}) for fixed k and n

Bound implies subgraphs of doubled Odd graphs are sparse for large cycles

Results lead to Ramsey-type implications for these graphs

Abstract

The generalized Tur\'{a}n number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of a graph $G.$ It is an important extension of the classical Tur\'{a}n number ${\rm ex}(n,H)$, 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 ${\rm ex}(J(n;k,k+1),C_{2r})$ with any fixed $k\in\mathbb{Z}^+$ and any $n\in\mathbb{Z}^+$ with $n\geq 2k+1.$ We also give an upper bound for ${\rm ex}(J(2k+1;k,k+1),C_{2r})$ with any $k\in\mathbb{Z}^+,$ where $J(2k+1;k,k+1)$ is known as doubled Odd graph $\widetilde{O}_{k+1}.$ This bound induces that the number of edges in any $C_{2r}$-free subgraph of $\widetilde{O}_{k+1}$ is $o(e(\widetilde{O}_{k+1}))$ for $r\geq 6,$ which also implies a Ramsey-type result.