Circular chromatic index of small graphs

TL;DR

This paper systematically determines the circular chromatic index of small graphs with maximum degree 4, 5, 6, constructing infinite families and refuting certain conjectures about their properties.

Contribution

It provides comprehensive values for small graphs' circular chromatic index and introduces new infinite graph families with specific index values, challenging existing conjectures.

Findings

Determined circular chromatic index for small graphs with degrees 4, 5, 6.

Constructed infinite families with indices in specific fractional sets.

Refuted variants of the Upper Gap Conjecture for edge-connected graphs.

Abstract

We systematically determine circular chromatic index of small graphs and multigraphs with maximum degree $4$, $5$, $6$ (and also their number for a given small order). We construct several infinite families of such graphs with circular chromatic index in the set $\{\Delta + 1/2, \Delta + 2/3, \Delta + 3/4$, $\Delta + 1\}$. Our results refute edge-connectivity variants of the ``Upper Gap Conjecture'' (about the non-existence of graphs with circular chromatic index just below $\Delta + 1$).