New bounds on even cycle creating Hamiltonian paths using expander graphs

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and K\"orner proved $$n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq n^{\frac{3}{4}n+o(n)}.$$ In this paper we close the superexponential gap between their lower and upper bounds by proving $$n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.$$ We also improve the previously established upper bounds on $H_n(C_{2k})$ for $k>3$, and we present a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri on the maximum number of so-called pairwise reversing permutations. One of our main tools is a theorem of Krivelevich, which roughly states that (certain kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised version incorporating suggestions by the referees (the changes are mainly in Section 5); v4: final version to appear in Combinatoric

New bounds on even cycle creating Hamiltonian paths using expander graphs

We say that two graphs on the same vertex set are $G$-creating if their union (the union of their edges) contains $G$ as a subgraph. Let $H_n(G)$ be the maximum number of pairwise $G$-creating Hamiltonian paths of $K_n$. Cohen, Fachini and K\"orner proved $$n^{\frac{1}{2}n-o(n)}\leq H_n(C_4) \leq n^{\frac{3}{4}n+o(n)}.$$ In this paper we close the superexponential gap between their lower and upper bounds by proving $$n^{\frac{1}{2}n-\frac{1}{2}\frac{n}{\log{n}}-O(1)}\leq H_n(C_4) \leq n^{\frac{1}{2}n+o\left(\frac{n}{\log{n}} \right)}.$$ We also improve the previously established upper bounds on $H_n(C_{2k})$ for $k>3$, and we present a small improvement on the lower bound of F\"uredi, Kantor, Monti and Sinaimeri on the maximum number of so-called pairwise reversing permutations. One of our main tools is a theorem of Krivelevich, which roughly states that (certain kinds of) good expanders contain many Hamiltonian paths.Comment: 14 pages, LaTeX2e; v2: updated Footnote 1 on Page 5; v3: revised version incorporating suggestions by the referees (the changes are mainly in Section 5); v4: final version to appear in Combinatoric

On k-neighbor separated permutations

Two permutations of [n]={1,2…n} are \textit{k-neighbor separated} if there are two elements that are neighbors in one of the permutations and that are separated by exactly k−2 other elements in the other permutation. Let the maximal number of pairwise k-neighbor separated permutations of [n] be denoted by P(n,k). In a previous paper, the authors have determined P(n,3) for every n, answering a question of Körner, Messuti and Simonyi affirmatively. In this paper we prove that for every fixed positive integer ℓ, P(n,2ℓ+1)=2n−o(n). We conjecture that for every fixed even k, P(n,k)=2n−o(n). We also show that this conjecture is asymptotically true in the following sense limk→∞limn→∞P(n,k)−−−−−−√n=2. Finally, we show that for even n, P(n,n)=3n/2

On reverse-free codes and permutations

A set F of ordered k-tuples of distinct elements of an n-set is pairwise reverse free if it does not contain two ordered k-tuples with the same pair of elements in the same pair of coordinates in reverse order. Let F(n, k) be the maximum size of a pairwise reverse-free set. In this paper we focus on the case of 3-tuples and prove limF(n, 3)/(n/3) = 5/4, more exactly, 5/14n 3 - 1/2n 2 - O(nlogn) > F(n, 3) ≥ 5/24 n 3 - 1/2n 2 + 5/8n, and here equality holds when n is a power of 3. Many problems remain open. © 2010 Societ y for Industrial and Applied Mathematics