Menu
A \( (k,d) \)-total coloring (\( k,d \in \mathbb{N}, k \geq 2d \)) of a graph \( G \) is an assignment \( c \) of colors \( \{0,1,\ldots,k-1\} \) to the vertices and edges of \( G \) such that \( d \leq |c(x_i) – c(x_j)| \leq k – d \) whenever \( x_i \) and \( x_j \) are two adjacent edges, two adjacent vertices, or an edge incident to a vertex. The circular total chromatic number \( \chi_c”(G) \) is defined by
\[\chi_c”(G) = \inf\{k/d : G \text{ has a } (k, d)\text{-total coloring}\}.\]
It was proved that \( \chi”(G) – 1 < \chi_c''(G) \leq \chi''(G) \) — where \( \chi''(G) \) is the total chromatic number of \( G \) — with equality for all type-1 graphs and most of the so far considered type-2 graphs. We determine an infinite class of graphs \( G \) such that \( \chi_c''(G) < \chi''(G) \) and we list all graphs of order \( <7 \) with this property.