On the total chromatic number of Steiner systems (Q2708239)

This paper conjectures that the total chromatic number of a Steiner system \(S(2,k,v)\), where \(3\leq k< v\), is equal to \(v\) and proves that this conjecture holds for projective planes, resolvable Steiner systems and cyclic Steiner systems by determining their total chromatic number. A relationship is established between this conjecture and the Erdős-Faber-Lovász conjecture which is expressed in [\textit{N. Hindman}, On a conjecture of Erdős, Faber, and Lovász about \(n\)-colourings, Can. J. Math. 33, 563-570 (1981; Zbl 0476.05009)] in the following form: if \(H\) is a linear hypergraph on \(n\) vertices with maximal vertex-degree at most \(n\), then the hyperedge chromatic number of \(H\) is also at most \(n\).