Unique Vertex Research Articles

On the chromatic index of almost all graphs

Vizing has shown that if G is a simple graph with maximum vertex-degree ϱ, then the chromatic index of G is either ϱ or ϱ + 1. In this note we prove that almost all graphs have a unique vertex of maximum degree, and we deduce that almost all graphs have chromatic index equal to their maximum degree. This settles a conjecture of the second author (in “Proceedings of the Fifth British Combinatorial Conference 1975”).

Antipodal Embeddings of Graphs

An antipodal graph D of diameter d has the property that each vertex υ has a unique (antipodal) vertex υ at distance d from υ in D. We show that any such D has circuits of length 2d passing through antipodal pairs of vertices. The identification of antipodal vertex-pairs in D produces a quotient graph G with a double cover projection morphism p : D→G. Using the two-fold quotient map of surfaces π : S2→RP2 where the real projective plane is obtained from the sphere, we study the relation between embeddings of a planar graph in S2 and embeddings of G in RP2. In particular, our main theorem establishes that every planar antipodal graph D has an embedding in S2 such that p is a restriction of the projection π.

Comparison of graphs by their number of spanning trees

The problem of comparison of graphs with the same number of vertices and edges by their number of spanning trees is considered. A set of operations on graphs which increase the number of their spanning trees is given. In particular, the following assertions are proved: (1) A disconnected graph is “better” (it destroys less trees if removed from the complete graph) than any connected separable graph whose blocks are components of the given graph. (2) The replacement, in a separable graph, of a block B with m edges which hangs at the unique vertex x, by the star with m edges fixed at its center in x, “worsens” the graph. (3) A chain (star) composed of identical and symmetric two-terminal networks is “better” (“worse”) than any other tree composed from the same two-terminal networks. (4) A chain (a circle) is “better” than any connected (respectively 2-connected) graph with the same number of edges. The article is concluded by a description of some operations on graphs which permit the extension of the results to a wider class of graphs.

Double covers of graphs

A projection morphism ρ: G1 → G2 of finite graphs maps the vertex-set of G1 onto the vertex-set of G2, and preserves adjacency. As an example, if each vertex v of the dodecahedron graph D is identified with its unique antipodal vertex v¯ (which has distance 5 from v) then this induces an identification of antipodal pairs of edges, and gives a (2:1)-projection p: D → P where P is the Petersen graph.In this paper a category-theoretical approach to graphs is used to define and study such double cover projections. An upper bound is found for the number of distinct double covers ρ: G1 → G2 for a given graph G2. A classification theorem for double cover projections is obtained, and it is shown that the n–dimensional octahedron graph K2,2,…,2 plays the role of universal object.

Automorphisms of subgraphs obtained by deleting a pendant vertex

A pendant vertex, x, of a finite graph, G, is ∗-fixed in case every automorphism of G − x fixes the unique vertex of G adjacent to x. It is proved that “almost all” finite graphs which have pendant vertices have ∗-fixed pendant vertices. In addition finite trees which have ∗-fixed pendant vertices are characterized.