Circular chromatic index of graphs of maximum degree 3

Abstract

This paper proves that if $G$ is a graph (parallel edges allowed) of maximum degree 3, then ${\chi }_c^{\prime }(G)\le 11/3$ provided that $G$ does not contain $H_1$ or $H_2$ as a subgraph, where $H_1$ and $H_2$ are obtained by subdividing one edge of $K_2^3$ (the graph with three parallel edges between two vertices) and $K_4$, respectively. As ${\chi }_c^{\prime }(H_1)={\chi }_c^{\prime }(H_2)=4$, our result implies that there is no graph $G$ with $11/3<{\chi }_c^{\prime }(G)<4$. It also implies that if $G$ is a 2-edge connected cubic graph, then ${\chi }^{\prime }(G)\le 11/3$.