2-弧着色有向图上的Euler问题和中国邮递员问题
The Euler and Chinese Postman Problems on 2-Arc-Colored Digraphs
摘要：
著名的中国邮递员问题（CPP）在无向图和有向图上都是多项式时间可解的。Gutin等人[Discrete Applied Math 217（2016）]通过证明$c$ -边着色图上的CPP对于每个$c\geq 2$都是多项式时间可解的来推广这些结果。在加权边着色图$G$上的CPP中，我们希望找到一个包含$G$的所有边的最小权适当着色的闭行走（如果每两个相邻的边是不同的颜色，则行走是适当着色的，包括闭行走中的最后一个和第一个边）。本文考虑弧着色有向图（适当着色的闭有向游动）上的CPP，并提供了加权2-弧着色有向图问题的多项式时间算法。这是一个有点令人惊讶的结果，因为决定2-弧着色有向图是否具有适当着色的有向圈是NP完全的[Gutin等人，离散数学191（1998）]。为了得到多项式时间算法，我们刻画了含有适当着色的欧拉轨迹的2-弧着色有向图。
英文摘要：
The famous Chinese Postman Problem (CPP) is polynomial time solvable on both undirected and directed graphs. Gutin et al. [Discrete Applied Math 217 (2016)] generalized these results by proving that CPP on $c$-edge-colored graphs is polynomial time solvable for every $c\geq 2$. In CPP on weighted edge-colored graphs $G$, we wish to find a minimum weight properly colored closed walk containing all edges of $G$ (a walk is properly colored if every two consecutive edges are of different color, including the last and first edges in a closed walk). In this paper, we consider CPP on arc-colored digraphs (for properly colored closed directed walks), and provide a polynomial-time algorithm for the problem on weighted 2-arc-colored digraphs. This is a somewhat surprising result since it is NP-complete to decide whether a 2-arc-colored digraph has a properly colored directed cycle [Gutin et al., Discrete Math 191 (1998)]. To obtain the polynomial-time algorithm, we characterize 2-arc-colored digraphs containing properly colored Euler trails.
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