Cooperative colorings of trees and of bipartite graphs

Given a system $(G_1, \ldots ,G_m)$ of graphs on the same vertex set $V$, a cooperative coloring is a choice of vertex sets $I_1, \ldots ,I_m$, such that $I_j$ is independent in $G_j$ and $\bigcup_{j=1}^{m}I_j = V$. For a class $\mathcal{G}$ of graphs, let $m_{\mathcal{G}}(d)$ be the minimal $m$ such that every $m$ graphs from $\mathcal{G}$ with maximum degree $d$ have a cooperative coloring. We prove that $\Omega(\log\log d) \le m_\mathcal{T}(d) \le O(\log d)$ and $\Omega(\log d)\le m_\mathcal{B}(d) \le O(d/\log d)$, where $\mathcal{T}$ is the class of trees and $\mathcal{B}$ is the class of bipartite graphs.Comment: 8 pages, 2 figures, accepted to the Electronic Journal of Combinatorics, corrections suggested by the referees have been incorporate

Bounded degree graphs and hypergraphs with no full rainbow matchings

Given a multi-hypergraph $G$ that is edge-colored into color classes $E_1, \ldots, E_n$, a full rainbow matching is a matching of $G$ that contains exactly one edge from each color class $E_i$. One way to guarantee the existence of a full rainbow matching is to have the size of each color class $E_i$ be sufficiently large compared to the maximum degree of $G$. In this paper, we apply a simple iterative method to construct edge-colored multi-hypergraphs with a given maximum degree, large color classes, and no full rainbow matchings. First, for every $r \ge 1$ and $\Delta \ge 2$, we construct edge-colored $r$-uniform multi-hypergraphs with maximum degree $\Delta$ such that each color class has size $|E_i| \ge r\Delta - 1$ and there is no full rainbow matching, which demonstrates that a theorem of Aharoni, Berger, and Meshulam (2005) is best possible. Second, we construct properly edge-colored multigraphs with no full rainbow matchings which disprove conjectures of Delcourt and Postle (2022). Finally, we apply results on full rainbow matchings to list edge-colorings and prove that a color degree generalization of Galvin's theorem (1995) does not hold