Hamilton paths with lasting separation

We determine the asymptotics of the largest cardinality of a set of Hamilton paths in the complete graph with vertex set [n] under the condition that for any two of the paths in the family there is a subpath of length k entirely contained in only one of them and edge{disjoint from the other one

Families of graph-different Hamilton paths

Let D be an arbitrary subset of the natural numbers. For every n, let M(n;D) be the maximum of the cardinality of a set of Hamiltonian paths in the complete graph K_n such that the union of any two paths from the family contains a not necessarily induced cycle of some length from D. We determine or bound the asymptotics of M(n;D) in various special cases. This problem is closely related to that of the permutation capacity of graphs and constitutes a further extension of the problem area around Shannon capacity. We also discuss how to generalize our cycle-difference problems and present an example where cycles are replaced by 4-cliques. These problems are in a natural duality to those of graph intersection, initiated by Erd\"os, Simonovits and S\'os. The lack of kernel structure as a natural candidate for optimum makes our problems quite challenging

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

Improved upper bounds on even-cycle creating Hamilton paths

We study the function $H_n(C_{2k})$, the maximum number of Hamilton paths such that the union of any pair of them contains $C_{2k}$ as a subgraph. We give upper bounds on this quantity for $k\in \{3, 4, 5\}$, improving results of Harcos and Solt\'esz, and we show that if a conjecture of Ustimenko is true then one additionally obtains improved upper bounds for all $k\geq 6$. In order to prove our results, we extend a theorem of Krivelevich which counts Hamilton cycles in $(n, d, \lambda)$-graphs to graphs which are not regular, and then apply this result to graphs constructed from polarity graphs of generalized polygons

Even cycle creating paths

We say that two graphs H1, H2 on the same vertex set are G-creating if the union of the two graphs contains G as a subgraph. Let H (n, k) be the maximum number of pairwise Ck-creating Hamiltonian paths of the complete graph Kn. The behavior of H (n, 2k + 1) is much better understood than the behavior of H (n, 2k), the former is an exponential function of n whereas the latter is larger than exponential, for every fixed k. We study H (n, k) for fixed k and n tending to infinity. The only nontrivial upper bound on H (n, 2k) was proved by Cohen, Fachini, and Körner in the case of k = 2: : (Formula presented.) In this paper, we generalize their method to prove that for every k ≥ 2, (Formula presented.) and a similar, slightly better upper bound holds when k is odd. Our proof uses constructions of bipartite, regular, C2k-free graphs with many edges given in papers by Reiman, Benson, Lazebnik, Ustimenko, and Woldar. © 2019 Wiley Periodicals, Inc

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