By Winter P. A.

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**Extra info for 2-3 graphs which have Vizings adjacency property**

**Example text**

Recall that a graph is Hamiltonian if it has a spanning cycle. If a cubic graph is Hamiltonian, we can draw it as a jV j-cycle with inserted chords, which leads to a convenient notation, the LCF notation, named for developers Lederberg, Coxeter, and Frucht. See [20, 33]. Given the Hamilton cycle, all we have to do to specify the graph is to list the lengths of chords measured in jumps when we traverse the vertices along the Hamilton cycle. Such a list is called the LCF notation. For instance, K4 can be described by Œ2; 2; 2; 2.

V1 [ V2 I k1 ; k2 /. The situation for constructibility changes drastically if we add as extra requirement that G must have girth larger than 4. This is not an arbitrary condition. A quadrilateral in the incidence graph corresponds to two distinct lines having two distinct points in common. k2 1/. k1 1/. Unfortunately, these obvious necessary conditions are not sufficient to ensure the existence of G. According to Gropp [38], there does not exist any 5-valent bipartite graph on 44 vertices of girth larger than 6.

To establish a lower bound on the number of vertices of a g-cage, we start with a single vertex, list its three neighbors, each of those has two neighbors, etc. For the case g D 2k C 1 up to the k-level, each neighbor set gives rise to two new unrecorded vertices, creating a binary tree whose root vertex is of valence 3; see Fig. 17. Edges between vertices on the same level are only allowed on level k, so we have at least 1 C 3 C 3 2 C 3 4 C C 3 2k 1 vertices, yielding the desired bound. 18. t u It turns out that the 6-cage is the Heawood graph; see Chap.