12 - Reductions of weights, coverages

Summary

For an eulerian weighted graph (G, w), if (G, w) is a contra pair, can we find another admissible eulerian weight w* of G such that (G, w*) remains as a contra pair while w*(G) < w(G) and 0 ≤ w*(e) ≤ w(e) for every edge e? If the answer to this question is “yes,” then we should concentrate on eulerian (1, 2)-weights in the study of contra pairs.

It is obvious that every bridgeless graph has a circuit cover. However, we do not know yet how “small” the maximum coverage would be. If one is able to find another circuit cover that reduces the coverage while the parity of coverage is retained, then one is able to reduce the coverage recursively down to 1 or 2, and the CDC conjecture is followed.

These two problems are both related to reductions: reduction of weight in a contra pair, and reduction of coverage of an existing cover. The first problem has a complete answer, and is studied in Section 12.1. The second problem, as we can see already, remains as an approach to the CDC conjecture (Section 12.2).

Note that reduction of total coverage without preserving the parity of coverage is the shortest cycle cover problem, which is discussed separately in Chapter 14.