关于有向图和$2$-边着色图到自反目标的非正常同态的存在性与不存在性
On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets
AI总结：
本文研究有向图和2-边着色图到自反目标的非正常同态，完全分类了树宽为2的情形，证明判定问题为NP完全，并为平面图定向色数问题提供新工具。
摘要：
我们考虑到自反有向图的非平凡同态，其中某些相邻顶点对具有相同的像。利用有向图的凸性概念，我们研究了那些不承认此类同态的有向图。我们完全分类了树宽为$2$且不承认此类同态的有向图，并证明判定一个图是否承认一个不承认此类同态的定向是NP完全的。我们对$2$-边着色图证明了类似的结果。我们将关于有向图的结果应用于平面图定向的色数研究——这是一个长期未解决的开放问题，并提供了新的工具。
英文摘要：
We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.
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