Circular total chromatic numbers of graphs

The circular total chromatic number   of a graph GG, written χc″(G), is the infimum rr permitting a function c:V(G)∪E(G)→[0,r) such that 1≤|c(x)−c(x′)|≤r−11≤|c(x)−c(x′)|≤r−1 whenever xx and x′x′ are two adjacent vertices, two adjacent edges, or an edge incident to a vertex. A real number rr is said to be realizable as the circular total chromatic number of graphs if there is a (possibly infinite) graph GG with χ″c(G)=rχ″c(G)=r. A natural question is which real numbers are realizable. It is known that r∈(3,4]r∈(3,4] is realizable if and only if r=3+1/kr=3+1/k for a positive integer kk. Very little was known about realizable r>4r>4. By determining the circular total chromatic numbers of some special graphs, it is shown in Ghebleh (2013) that for each integer n≥3n≥3, there is a strictly decreasing sequence of realizable reals approaching nn. However, it remained an open question as whether there is a bounded infinite strictly increasing sequence of realizable reals. In this paper, we answer this question is affirmative and prove that for any integer n≥4n≥4, for rational ϵ∈(0,1/3)ϵ∈(0,1/3), there is a finite simple graph GG with χ″c(G)=n+1+ϵχ″c(G)=n+1+ϵ.