On the Set of Circular Total Chromatic Numbers of Graphs

TL;DR

This paper demonstrates that for any integer degree, the set of circular total chromatic numbers of graphs is dense around that degree, and infinitely large, including bipartite graphs.

Contribution

It constructs graphs with maximum degree r-1 having circular total chromatic numbers arbitrarily close to r, proving the set's density and infinitude.

Findings

Every integer r ≥ 3 is an accumulation point.

The set of circular total chromatic numbers is infinite for each maximum degree.

Results hold for bipartite graphs as well.

Abstract

For every integer $r\ge3$ and every $\eps>0$ we construct a graph with maximum degree $r-1$ whose circular total chromatic number is in the interval $(r,r+\eps)$. This proves that (i) every integer $r\ge3$ is an accumulation point of the set of circular total chromatic numbers of graphs, and (ii) for every $\Delta\ge2$, the set of circular total chromatic numbers of graphs with maximum degree $\Delta$ is infinite. All these results hold for the set of circular total chromatic numbers of bipartite graphs as well.