Colouring of Voloshin for ATS(v)

• Main Author: Alberto Amato
• Format: Article
• Language: English
• Published: Università degli Studi di Catania 2002-11-01
• Series: Le Matematiche
• Online Access: http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/213

A mixed hypergraph is a triple <em>H=(S,C,D)</em>, where <em>S</em> is the vertex set and each of <em>C,D</em> is a family of not-empty subsets of <em>S</em>, the <em>C</em>-edges and <em>D</em>-edges respectively. A strict <em>k</em>-colouring of <em>H</em> is a surjection <em>f</em>  from the vertex set into a set of colours <em>{1, 2, . . . , k}</em> so that each <em>C</em>-edge contains at least two distinct vertices <em>x, y</em> such that <em>f(x) = f(y)</em> and each <em>D</em>-edge contains at least two vertices <em>x, y</em> such that <em>f(x)=f(y)</em>. For each <em>k ∈ {1, 2, . . . , |S|}</em>, let <em>r_k</em> be the number of partitions of the vertex set into<em> k</em> not-empty parts (the colour classes) such that the colouring constraint is satisfied on each <em>C</em>-edge and <em>D</em>-edge. The vector <em>R(H ) = (r_1 , . . . , r_k )</em> is called the chromatic spectrum of <em>H</em>. These concepts were introduced by V. Voloshin in 1993 [6].<br /><br />In this paper we examine colourings of mixed hypergraphs in the case that H is an ATS(v).<br />