Can Colour-Blind Distinguish Colour Palettes?

Let c : E(G) → [k] be a colouring, not necessarily proper, of edges of a graph G. For a vertex v ∈ V , let c(v) = (a1, . . . , ak), where ai = |{u : uv ∈ E(G), c(uv) = i}|, for i ∈ [k]. If we re-order the sequence c(v) non-decreasingly, we obtain a sequence c∗(v) = (d1, . . . , dk), called a palette of a vertex v. This can be viewed as the most comprehensive information about colours incident with v which can be delivered by a person who is unable to name colours but distinguishes one from another. The smallest k for which there exists a c such that c∗ is a proper colouring of vertices of G is called the colour-blind index of a graph G, and is denoted by dal(G). We conjecture that there is a constant K such that dal(G) 6 K for every graph G for which the parameter is well defined. As our main result we prove that K 6 6 for regular graphs of sufficiently large degree, and for irregular graphs with δ(G) and ∆(G) satisfying certain conditions. The proofs are based on the Lopsided Lovász Local Lemma. We also show that K = 3 for all regular bipartite graphs, and for complete graphs of order n > 8.