Recently, Behr introduced a notion of the chromatic index of signed graphs and proved that for every signed graph $(G$, $\sigma)$ it holds that \[ \Delta(G)\leq\chi'(G\text{, }\sigma)\leq\Delta(G)+1\text{,} \] where $\Delta(G)$ is the maximum degree of $G$ and $\chi'$ denotes its chromatic index. In general, the chromatic index of $(G$, $\sigma)$ depends on both the underlying graph $G$ and the signature $\sigma$. In the paper we study graphs $G$ for which $\chi'(G$, $\sigma)$ does not depend on $\sigma$. To this aim we introduce two new classes of graphs, namely $1^\pm$ and $2^\pm$, such that graph $G$ is of class $1^\pm$ (respectively, $2^\pm$) if and only if $\chi'(G$, $\sigma)=\Delta(G)$ (respectively, $\chi'(G$, $\sigma)=\Delta(G)+1$) for all possible signatures $\sigma$. We prove that all wheels, necklaces, complete bipartite graphs $K_{r,t}$ with $r\neq t$ and almost all cacti graphs are of class $1^\pm$. Moreover, we give sufficient and necessary conditions for a graph to be of class $2^\pm$, i.e. we show that these graphs must have odd maximum degree and give examples of such graphs with arbitrary odd maximum degree bigger that $1$.

翻译：最近，Behr引入了符号图染色指数的概念，并证明了对于每个符号图$(G,\sigma)$，有 \[ \Delta(G)\leq\chi'(G,\sigma)\leq\Delta(G)+1, \] 其中$\Delta(G)$是$G$的最大度数，$\chi'$表示其染色指数。通常，$(G,\sigma)$的染色指数既依赖于基础图$G$也依赖于符号函数$\sigma$。本文研究使得$\chi'(G,\sigma)$不依赖于$\sigma$的图$G$。为此，我们引入了两个新图类，即$1^\pm$类和$2^\pm$类，使得图$G$属于$1^\pm$类（分别地，$2^\pm$类）当且仅当对所有可能的符号函数$\sigma$有$\chi'(G,\sigma)=\Delta(G)$（分别地，$\chi'(G,\sigma)=\Delta(G)+1$）。我们证明了所有轮图、项链图、$r\neq t$的完全二部图$K_{r,t}$以及几乎所有仙人掌图都属于$1^\pm$类。此外，我们给出了图属于$2^\pm$类的充分必要条件，即证明了这类图必须具有奇数最大度数，并给出了具有任意大于1的奇数最大度数的此类图实例。