A signed graph is a graph together with an assignment of signs to the edges. A closed walk in a signed graph is said to be positive (negative) if it has an even (odd) number of negative edges, counting repetition. Recognizing the signs of closed walks as one of the key structural properties of a signed graph, we define a homomorphism of a signed graph $(G,\sigma )$ to a signed graph $(H,\pi )$ to be a mapping of vertices and edges of $G$ to (respectively) vertices and edges of $H$ which preserves incidence, adjacency and the signs of closed walks.

In this work we first give a characterization of the sets of closed walks in a graph $G$ that correspond to the set of negative closed walks in some signed graph on $G$. We also give an easy algorithm for the corresponding decision problem.

After verifying the equivalence between this definition and earlier ones, we discuss the relation between homomorphisms of signed graphs and those of 2-edge-colored graphs. Next we provide some basic no-homomorphism lemmas. These lemmas lead to a general method of defining chromatic number which is discussed at length. Finally, we list a few problems that are the driving force behind the study of homomorphisms of signed graphs.

有符号图是一个图形，其符号与边的分配相同。如果签名图中的闭合行走具有负数的偶数（奇数个）（计算重复次数），则称其为正（负）。认识到封闭步行的迹象是签名图的关键结构特性之一，我们定义了签名图的同构$（G，\sigma ）$ 到签名图 $（H，\pi ）$ 成为顶点和边的映射 $G$ （分别）到的顶点和边缘 $H$ 保留了发生率，邻接关系和封闭步行的迹象。

在这项工作中，我们首先对图中的封闭步道进行表征 $G$ 对应于某个有向图中的负闭合行走的集合 $G$。我们还为相应的决策问题提供了一种简单的算法。

在验证了该定义与早期定义之间的等价性之后，我们讨论了带符号图的同构与2边色图的同构之间的关系。接下来，我们提供一些基本的非同态引理。这些引理导致了定义色数的通用方法，该方法将详细讨论。最后，我们列出了一些问题，这些问题是有符号图的同态性研究背后的驱动力。