The Interleaved Multichromatic Number of a Graph

For k ≥ 1, we consider interleaved k-tuple colorings of the nodes of a graph, that is, assignments of k distinct natural numbers to each node in such a way that nodes that are connected by an edge receive numbers that are strictly alternating between them with respect to the relation <. If it takes at least χint(G) distinct numbers to provide graph G with such a coloring, then the interleaved multichromatic number of G is χ∗int(G) = infk≥1 χ k int(G)/k and is known to be given by a function of the simple cycles of G under acyclic orientations if G is connected. This paper contains a new proof of this result. Unlike the original proof, the new proof makes no assumptions on the connectedness of G, nor does it resort to the possible applications of interleaved k-tuple colorings and their properties.