A digraph is called ubiquitous if every digraph that contains arbitrarily many vertex-disjoint copies of also contains infinitely many vertex-disjoint copies of . We study oriented double rays, that is, digraphs whose underlying undirected graphs are double rays. Calling a vertex of an oriented double ray a turn if it has in-degree or out-degree 2, we prove that an oriented double ray with at least one turn is ubiquitous if and only if it has a (finite) odd number of turns. It remains an open problem to determine whether the consistently oriented double ray is ubiquitous.
Applications of graph colouring often involve taking restrictions into account, and it is desirable to have multiple (disjoint) solutions. In the optimal case, where there is a partition into disjoint colourings, we speak of a packing. However, even for complete bipartite graphs, the list chromatic number can be arbitrarily large, and its exact determination is generally difficult. For the packing variant, this question becomes even harder. In this paper, we study the correspondence- and list-packing numbers of (asymmetric) complete bipartite graphs. In the most asymmetric cases, Latin squares come into play. Our results show that every can be equal to the correspondence packing number of a graph. We disprove a recent conjecture that relates the list packing number and the list flexibility number. Additionally, we improve the threshold functions for the correspondence packing variant.
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