Rainbow Paths with Prescribed Ends

It was conjectured in [S. Akbari, F. Khaghanpoor, and S. Moazzeni. Colorful paths in vertex coloring of graphs. Preprint] that, if $G$ is a connected graph distinct from $C_7$, then there is a $\chi(G)$-coloring of $G$ in which every vertex $v\in V(G)$ is an initial vertex of a path $P$ with $\chi(G)$ vertices whose colors are different. In [S. Akbari, V. Liaghat, and A. Nikzad. Colorful paths in vertex coloring of graphs. Electron. J. Combin. 18(1): P17, 9pp, 2011] this was proved with $\lfloor\frac{\chi(G)}{2} \rfloor $ vertices instead of $\chi(G)$ vertices. We strengthen this to $\chi(G)-1$ vertices. We also prove that every connected graph with at least one edge has a proper $k$-coloring (for some $k$) such that every vertex of color $i$ has a neighbor of color $i+1$ (mod $k$). $C_5$ shows that $k$ may have to be greater than the chromatic number. However, if the graph is connected, infinite and locally finite, and has finite chromatic number, then the $k$-coloring exists for every $k \geq \chi(G)$. In fact, the $k$-coloring can be chosen such that every vertex is a starting vertex of an infinite path such that the color increases by $1$ (mod $k$) along each edge. The method is based on the circular chromatic number $\chi_c(G)$. In particular, we verify the above conjecture for all connected graphs whose circular chromatic number equals the chromatic number.