On Panchromatic Digraphs and the Panchromatic Number
Tóm tắt
Let D = (V, A) be a simple finite digraph, and let
$${\pi(D)}$$
, the panchromatic number of D, be the maximum number of colours k such that for each (effective, or onto) colouring of its arcs
$${\varsigma : A \to [k]}$$
a monochromatic path kernel
N ⊂ V, as introduced in Galeana-Sánchez (Discrete Math 184: 87–99, 1998), exists. It is not hard to see that D has a kernel—in the sense of Von Neumann and Morgenstern (Theory of games and economic behaviour. Princeton University Press, Princeton, 1944)—if and only if
$${\pi(D) = |A|}$$
. In this note this invariant is introduced and some of its structural bounds are studied. For example, the celebrated theorem of Sands et al. (J Comb Theory Ser 33: 271–275, 1982), in terms of this invariant, settles that
$${\pi(D) \geq 2}$$
. It will be proved that
$$\pi(D) < |A| \iff \pi(D) < \min \left\{2\sqrt{\chi(D)}, \chi(L(D)), \theta(D) + \max {\text d}_c(K_i) + 1\right\},$$
where
$${\chi(\cdot)}$$
denotes the chromatic number,
$${L(\cdot)}$$
denotes the line digraph,
$${\theta(\cdot)}$$
denotes the minimum partition into complete graphs of the underlying graph and
$${\text{d}_c(\cdot)}$$
denotes the dichromatic number. We also introduce the notion of a panchromatic digraph which is a digraph D such that for every k ≤ |A| and every k-colouring of its arcs, it has a monochromatic path kernel. Some classes of panchromatic digraphs are further characterised.
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