ON INCIDENCE COLORING OF SIGNED GRAPHS

An incidence of a graph $G$ is a pair $(x,e)$, where $x$ is a vertex of $G$ and $e$ is an edge of $G$ incident to $x$. Two incidences $(x,e)$ and $(y,f)$ are adjacent if any one of the following holds: (i) $x=y$, or (ii) $e=f$, or (iii) $xy=e$ or $f$. In [3], Brualdi and Massey introduced the concept of incidence coloring of a graph $G$ as a mapping from the set of incidences of $G$ to a finite set of colors such that adjacent incidences receive distinct colors. A signed graph $(G,\sigma)$ consists of a graph $G$ and the signature $\sigma : E(G)\rightarrow \{+1,-1\}$. In this paper, we define an incidence coloring of signed graphs as a natural generalization of the usual notion of incidence coloring of unsigned graphs. We prove that our definition is compatible with switching operation. We also prove that the incidence chromatic number (in signed sense) of a signed graph $(G,\sigma)$ coincide with the incidence chromatic number (in the usual unsigned sense) of its underlying graph $G$. The exact value or upper bounds which are known for the incidence chromatic numbers of some well-known families of unsigned graphs are also mentioned for their signed versions, namely, signed cycles, signed trees, signed complete graphs and signed toroidal grids.