On the set of circular total chromatic numbers of graphs

While total coloring of graphs and circular coloring of graphs have received a great deal of attention from researchers, there has been a limited amount of work done on circular total colorings. In particular, only a finite number of values a>4 have been realized as the circular total chromatic number of any graph, and it is not yet known whether the set of circular total chromatic numbers of graphs with a given maximum degree r⩾3 is finite or infinite. In this paper, we construct for every integer r⩾3 and every ε>0, a graph with maximum degree r−1 whose circular total chromatic number is in the interval (r,r+ε). This proves that (i) every integer r⩾3 is an accumulation point of the set of circular total chromatic numbers of graphs, and (ii) for every Δ⩾2, the set of circular total chromatic numbers of graphs with maximum degree Δ is infinite. These results hold also for the set of circular total chromatic numbers of bipartite graphs.