Auswahl der wissenschaftlichen Literatur zum Thema „Rainbow Hamilton cycle“

The anti-Ramsey number ARG,H is the maximum number of colors in an edge-coloring of G such that G contains no rainbow subgraphs isomorphic to H. In this paper, we discuss the anti-Ramsey numbers ARKp1,p2,…,pk,Tn, ARKp1,p2,…,pk,ℳ, and ARKp1,p2,…,pk,C of Kp1,p2,…,pk, where Tn,ℳ, and C denote the family of all spanning trees, the family of all perfect matchings, and the family of all Hamilton cycles in Kp1,p2,…,pk, respectively.

Let $H_{n,p,r}^{(k)}$ denote a randomly colored random hypergraph, constructed on the vertex set $[n]$ by taking each $k$-tuple independently with probability $p$, and then independently coloring it with a random color from the set $[r]$. Let $H$ be a $k$-uniform hypergraph of order $n$. An $\ell$-Hamilton cycle is a spanning subhypergraph $C$ of $H$ with $n/(k-\ell)$ edges and such that for some cyclic ordering of the vertices each edge of $C$ consists of $k$ consecutive vertices and every pair of adjacent edges in $C$ intersects in precisely $\ell$ vertices.In this note we study the existence of rainbow $\ell$-Hamilton cycles (that is every edge receives a different color) in $H_{n,p,r}^{(k)}$. We mainly focus on the most restrictive case when $r = n/(k-\ell)$. In particular, we show that for the so called tight Hamilton cycles ($\ell=k-1$) $p = e^2/n$ is the sharp threshold for the existence of a rainbow tight Hamilton cycle in $H_{n,p,n}^{(k)}$ for each $k\ge 4$.

Let $K_n^{(k)}$ be the complete $k$-uniform hypergraph, $k\ge3$, and let $\ell$ be an integer such that $1\le \ell\le k-1$ and $k-\ell$ divides $n$. An $\ell$-overlapping Hamilton cycle in $K_n^{(k)}$ is a spanning subhypergraph $C$ of $K_n^{(k)}$ with $n/(k-\ell)$ edges and such that for some cyclic ordering of the vertices each edge of $C$ consists of $k$ consecutive vertices and every pair of adjacent edges in $C$ intersects in precisely $\ell$ vertices.We show that, for some constant $c=c(k,\ell)$ and sufficiently large $n$, for every coloring (partition) of the edges of $K_n^{(k)}$ which uses arbitrarily many colors but no color appears more than $cn^{k-\ell}$ times, there exists a rainbow $\ell$-overlapping Hamilton cycle $C$, that is every edge of $C$ receives a different color. We also prove that, for some constant $c'=c'(k,\ell)$ and sufficiently large $n$, for every coloring of the edges of $K_n^{(k)}$ in which the maximum degree of the subhypergraph induced by any single color is bounded by $c'n^{k-\ell}$, there exists a properly colored $\ell$-overlapping Hamilton cycle $C$, that is every two adjacent edges receive different colors. For $\ell=1$, both results are (trivially) best possible up to the constants. It is an open question if our results are also optimal for $2\le\ell\le k-1$.The proofs rely on a version of the Lovász Local Lemma and incorporate some ideas from Albert, Frieze, and Reed.

La combinatoire extrémale est l'une des branches les plus vigoureuses des mathématiques combinatoires au cours des dernières décennies, et elle a été largement utilisée en informatique, en conception de réseaux et en conception de codage. Elle se concentre sur la détermination de la taille maximale ou minimale possible de certaines structures combinatoires, sous certaines conditions ou contraintes. Les ensembles hôtes peuvent être des graphes, des digraphes, des graphes aléatoires, des hypergraphes, des entiers, des nombres premiers, des ensembles, des graphes avec arêtes colorées, etc. Les structures locales peuvent être des appariements, des cliques, des cycles, des arbres, des sous-graphes couvrants (facteurs F, cycles Hamiltoniens), des familles d'intersection, des progressions arithmétiques, des solutions pour certaines équations (par exemple, x+y=z), des sous-graphes arc-en-ciel, etc. En particulier, la théorie des graphes extrémaux est une branche importante de la combinatoire extrémale, qui traite principalement de la manière dont les propriétés générales d'un graphe contrôlent la structure locale du graphe. Nous étudions l'existence d'un cycle Hamiltonien rainbow dans les systèmes de k-graphes, l'existence d'un appariement parfait rainbow dans les systèmes de k-graphes et l'existence d'un cycle long arc-en-ciel dans des graphes correctement colorés par les arêtes
Extremal Combinatorics is one of the most vigorous branch of Combinatorial Mathematics in recent decades and it has been widely used in Computer Science, Network Design and Coding Design. It focuses on determining the maximum or minimum possible size of certain combinatorial structures, subject to certain conditions or constraints. The host sets could be graphs, digraphs, random graphs, hypergraphs, integers, primes, sets, edge-colored graphs and so on. The local structures could be matchings, cliques, cycles, trees, spanning subgraphs (F-factors, Hamilton cycles), intersecting families, arithmetic progressions, solutions for some equations (e.g. x₊y₌z), rainbow subgraphs and so on. In particular, Extremal Graph Theory is a significant branch of Extremal Combinatorics, which primarily explores how the overall properties of a graph influence its local structures. We study the existence of a rainbow Hamilton cycle in k-graph systems, the existence of rainbow perfect matching in k-graph systems, and the existence of long rainbow cycle in properly edge-colored graphs