On Panchromatic Digraphs and the Panchromatic Number

Let D = (V,A) be a simple digraph, and let π(D), the panchromatic number of D, be the maximum number of colours k such that for each colouring of its arcs ς : A → [k] a kernel by monochromatic paths N ⊂ V , as introduced in [6], exists. It is not hard to see that D has a kernel —in the sense of Von Neumann [11]— if and only if π(D) = |A|. In this note this invariant is introduced and some of its structural bounds are studied. For example, the celebrated theorem of Sands et al. [?], in terms of this invariant, settles that π(D) ≥ 2. It will be proved that π(D) < |A| ⇐⇒ π(D) < min { 2 √ χ(D), χ(L(D)), θ(D) + max dc(Ki) + 1 }