Paths, chains, and antipaths

Abstract

Some variations on Eulerian partitions of edge sets and arc sets are discussed. We consider partitions into (odd) paths, chains and antipaths; these are paths where every second arc is reversed. Packing problems are also examined and min-rnax results are derived by using network flow techniques.

On some properties of suboptimal colorings of graphs

“…Furthermore, it also holds for q Ն ((G))/ 2. This can be seen easily, if we notice that in any coloring there exists a chain of (G) nodes where no two have the same color [3]. Taking the central node of such a chain for v gives the result.…”

“…We also say that N q (v) is the set of nodes which are at distance at most q from v. Optimal colorings of graphs G [i.e., (G)-colorings] have been studied rather extensively. They have many remarkable properties, and let us mention the following [3]: In any (G)-coloring of a graph G and for any permutation a 1 , a 2 , . .…”

DNA-templated synthesis optimization

“…The approach will employ so-called antipaths [5], which is a sequence of adjacent edges in a digraph, where every visited edge has opposite direction of the previously visited edge; and we will need some further restrictions defined below.…”

“…Constraint (2) makes sure that each place holder is assigned a tag. Constraints (5), (6), and (7) implement the interference information, where the latter two use a standard formulation for ensuring that one out of two constraints hold. The remaining constraints define domains.…”