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 $\chi_c'(G) \leq 11/3$ 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 $\chi_c'(H_1) = \chi_c'(H_2) = 4$, our result implies that there is no graph $G$ with $11/3 < \chi_c'(G) < 4$. It also implies that if $G$ is a 2-edge connected cubic graph, then $\chi'(G) \le 11/3$.