Generalized Fractional and Circular Total Colorings of Graphs

• Main Authors: Kemnitz Arnfried, Marangio Massimiliano, Mihók Peter, Oravcová Janka, Soták Roman
• Format: Article
• Language: English
• Published: Sciendo 2015-08-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: graph property; (p,q)-total coloring; circular coloring; fractional coloring; fractional (p,q)-total chromatic number; circular (p,q)- total chromatic number.
• Online Access: https://doi.org/10.7151/dmgt.1812

Let P and Q be additive and hereditary graph properties, r, s ∈ N, r ≥ s, and [ℤr]s be the set of all s-element subsets of ℤr. An (r, s)-fractional (P,Q)-total coloring of G is an assignment h : V (G) ∪ E(G) → [ℤr]s such that for each i ∈ ℤr the following holds: the vertices of G whose color sets contain color i induce a subgraph of G with property P, edges with color sets containing color i induce a subgraph of G with property Q, and the color sets of incident vertices and edges are disjoint. If each vertex and edge of G is colored with a set of s consecutive elements of ℤr we obtain an (r, s)-circular (P,Q)-total coloring of G. In this paper we present basic results on (r, s)-fractional/circular (P,Q)-total colorings. We introduce the fractional and circular (P,Q)-total chromatic number of a graph and we determine this number for complete graphs and some classes of additive and hereditary properties.