We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello, Fellows, and Stillwell asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show some solvable cases and, in particular, completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding $k$-tuples of pairwise disjoint perfect matchings in regular graphs).
Interval non-edge-colorable bipartite graphs and multigraphs
一个图$G$的边染色称为区间$t$ -染色，如果所有的颜色都被使用，并且关联到$G$的任何顶点的边的颜色都是不同的，并且形成一个整数区间。1991年Erdens构造了一个顶点数为27，最大度为13的无区间着色的二部图. Erd的反例是已知的最小（在最大度意义下）的非区间可着色的二部图. 1992年，Hansen证明了所有最大度不超过3的二部图都有区间着色。本文给出了区间非边着色二部图的一些构造方法。特别地，利用这些方法，我们构造了三个无区间着色的二部图，它们分别含有20，19，21个顶点，最大度分别为11，12，13.这部分地回答了[T. R.詹森湾Toft，Graph coloring problems，Wiley Interscience Series in Discrete Mathematics and Optimization，1995年，第204页]。我们也考虑类似的问题，二部多重图。
