Graphs and subgraphs. II

Abstract

title. The terminology is unchanged and the enumeration from the first paper is continued. We suppose as before that the basic graph G is finite and without loops. We observe that by small reformulations in certain statements loops could have been included in the theory, while it is essential for several results that G be finite. Our starting point in Chapter 4 is the theorem of Petersen about the interrelation between conformal subgraphs (subgraphs with the same local degrees). The choice available in the determination of the edges in the desired subgraph H leads to the concept of free equivalence as well as to a unique decomposition of the graph into a bound and a free part. Criteria are established to determine when an edge is free or bound. These are applied, in particular, to the subgraphs with constant proportions for the local degrees. The existence of such subgraphs was established in Chapter 3. Here it is shown that for these all edges are free equivalent; hence the same is true for the regular graphs and subgraphs discussed in ?3.2. A special case is a well known result by Petersen for subgraphs of first degree in regular graphs of degree 3 without peninsulas. It is of interest to note that this particular theorem has an important application for the method of alternating paths in general graph theory. In ?4.4 it is shown that the accessible characters of vertices under alternating H-paths is invariant, that is, do not depend on H but only upon the class of conformal subgraphs to which H belongs. In Chapter 5 the concept of free equivalence is discussed in greater detail. Its relation to the so-called cursal equivalence is examined. Among the results are criteria for two vertices to have the same accessible set with identical cursal properties. There exist a considerable number of problems related to those analysed, but these may be left to others. Chapter 6 contains observations on regular graphs which are completely decomposable, that is, are the sum of subgraphs of first degree.