Interval cyclic edge-colorings of graphs

Abstract

A proper edge-coloring of a graph $G$ with colors $1,\dots ,t$ is called an \emph{interval cyclic $t$-coloring} if all colors are used, and the edges incident to each vertex $v\in V(G)$ are colored by $d_G(v)$ consecutive colors modulo $t$, where $d_G(v)$ is the degree of a vertex $v$ in $G$. A graph $G$ is \emph{interval cyclically colorable} if it has an interval cyclic $t$-coloring for some positive integer $t$. The set of all interval cyclically colorable graphs is denoted by ${\mathfrak{N}}_c$. For a graph $G\in {\mathfrak{N}}_c$, the least and the greatest values of $t$ for which it has an interval cyclic $t$-coloring are denoted by $w_c(G)$ and $W_c(G)$, respectively. In this paper we investigate some properties of interval cyclic colorings. In particular, we prove that if $G$ is a triangle-free graph with at least two vertices and $G\in {\mathfrak{N}}_c$, then $W_c(G)\le |V(G)|+\Delta (G)-2$. We also obtain bounds on $w_c(G)$ and $W_c(G)$ for various classes of graphs. Finally, we give some methods for constructing of interval cyclically non-colorable graphs.

arXiv comment23 pages, 5 figures