Rainbow matchings in Dirac bipartite graphs

This is the peer reviewed version of the following article: Coulson, M, Perarnau, G. Rainbow matchings in Dirac bipartite graphs. Random Struct Alg. 2019; 55: 271– 289., which has been published in final form at https://doi.org/10.1002/rsa.20835. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived VersionsWe show the existence of rainbow perfect matchings in µn-bounded edge colorings of Dirac bipartite graphs, for a sufficiently small µ¿>¿0. As an application of our results, we obtain several results on the existence of rainbow k-factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree

Rainbow Hamilton cycle in hypergraph systems

R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di proved that every $n$-vertex $k$-graph $H$, $k\geq3, \gamma>0$ and $n$ is sufficiently large, with $\delta_{k-1}(H)\geq(1/2+\gamma)n$ contains a tight Hamilton cycle, which can be seen as a generalization of Dirac's theorem in hypergraphs. In this paper, we extend this result to the rainbow setting as follows. A $k$-graph system $\textbf{H}=\{H_i\}_{i\in[m]}$ is a family of not necessarily distinct $k$-graphs on the same $n$-vertex set $V$, a $k$-graph $G$ on $V$ is rainbow if $E(G)\subseteq\bigcup_{i\in[m]}E(H_i)$ and $|E(G)\cap E(H_i)|\leq 1$ for $i\in[m]$. Then we show that given $k\geq3, \gamma>0$, sufficiently large $n$ and an $n$-vertex $k$-graph system $\textbf{H}=\{H_i\}_{i\in[n]}$, if $\delta_{k-1}(H_i)\geq(1/2+\gamma)n$ for $i\in[n]$, then there exists a rainbow tight Hamilton cycle.Comment: 20 pages,5 figure

Robust Hamiltonicity in families of Dirac graphs

A graph is called Dirac if its minimum degree is at least half of the number of vertices in it. Joos and Kim showed that every collection $\mathbb{G}=\{G_1,\ldots,G_n\}$ of Dirac graphs on the same vertex set $V$ of size $n$ contains a Hamilton cycle transversal, i.e., a Hamilton cycle $H$ on $V$ with a bijection $\phi:E(H)\rightarrow [n]$ such that $e\in G_{\phi(e)}$ for every $e\in E(H)$. In this paper, we determine up to a multiplicative constant, the threshold for the existence of a Hamilton cycle transversal in a collection of random subgraphs of Dirac graphs in various settings. Our proofs rely on constructing a spread measure on the set of Hamilton cycle transversals of a family of Dirac graphs. As a corollary, we obtain that every collection of $n$ Dirac graphs on $n$ vertices contains at least $(cn)^{2n}$ different Hamilton cycle transversals $(H,\phi)$ for some absolute constant $c>0$. This is optimal up to the constant $c$. Finally, we show that if $n$ is sufficiently large, then every such collection spans $n/2$ pairwise edge-disjoint Hamilton cycle transversals, and this is best possible. These statements generalize classical counting results of Hamilton cycles in a single Dirac graph

Rainbow Perfect and Near-Perfect Matchings in Complete Graphs with Edges Colored by Circular Distance

Given an edge-colored complete graph Kn on n vertices, a perfect (respectively, near-perfect) matching M in Kn with an even (respectively, odd) number of vertices is rainbow if all edges have distinct colors. In this paper, we consider an edge coloring of Kn by circular distance, and we denote the resulting complete graph by K●n. We show that when K●n has an even number of vertices, it contains a rainbow perfect matching if and only if n=8k or n=8k+2, where k is a nonnegative integer. In the case of an odd number of vertices, Kirkman matching is known to be a rainbow near-perfect matching in K●n. However, real-world applications sometimes require multiple rainbow near-perfect matchings. We propose a method for using a recursive algorithm to generate multiple rainbow near-perfect matchings in K●n

Universality for transversal Hamilton cycles

Let $\mathbf{G}=\{G_1, \ldots, G_m\}$ be a graph collection on a common vertex set $V$ of size $n$ such that $\delta(G_i) \geq (1+o(1))n/2$ for every $i \in [m]$. We show that $\mathbf{G}$ contains every Hamilton cycle pattern. That is, for every map $\chi: [n] \to [m]$ there is a Hamilton cycle whose $i$-th edge lies in $G_{\chi(i)}$.Comment: 18 page