By Sergei Ovchinnikov

This introductory textual content in graph idea makes a speciality of partial cubes, that are graphs which are isometrically embeddable into hypercubes of an arbitrary size, in addition to bipartite graphs, and cubical graphs. This department of graph conception has constructed swiftly in the past 3 a long time, generating fascinating effects and setting up hyperlinks to different branches of mathematics.

Currently, Graphs and Cubes is the one publication available to buy that provides a accomplished insurance of cubical graph and partial dice theories. Many workouts, in addition to historic notes, are integrated on the finish of each bankruptcy, and readers are inspired to discover the routines absolutely, and use them as a foundation for study projects.

The necessities for this article contain familiarity with simple mathematical techniques and strategies at the point of undergraduate classes in discrete arithmetic, linear algebra, workforce idea, and topology of Euclidean areas. whereas the e-book is meant for lower-division graduate scholars in arithmetic, it will likely be of curiosity to a wider viewers; due to their wealthy structural houses, partial cubes seem in theoretical laptop technological know-how, coding thought, genetics, or even the political and social sciences.

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**Example text**

In this language, Hall’s Theorem tells us that A has an SDR if and only if | ∪i∈J Ai | ≥ |J| for all subsets J of I. In this form it appeared in Hall’s original paper (1935) where it was used to answer a question in group theory. It is hard to underestimate the importance of matching theory in both pure and applied mathematics. For a good account of the history and methods of the theory the reader is referred to Lov´asz and Plummer (1986). 1. Show that in a finite graph the number of vertices of odd degree is even.

14). 18. A connected graph G = (V, E) is bipartite if and only if the semicubes Wab and Wba form a partition of the vertex set V for every edge ab ∈ E, that is, Wab ∩ Wba = ∅ and Wab ∪ Wba = V, for all ab ∈ E. 9 where the graphs K2,3 and C6 are bipartite and C5 is not. By definition, d(x, a) < d(x, b) for x ∈ Wab . In fact, one can say more about the relation between these two distances. 4, we have the following result. 19. Let x ∈ Wab for some edge ab of a connected graph. Then d(x, b) = d(x, a) + 1.

The graph T −v does not contain cycles because there are no cycles in T . It follows that T − v is a tree. Clearly, there are |V | − 1 vertices and |E| − 1 edges in the tree T − v. Finite trees can be characterized as follows. 32. 7) that is, the number of edges of G is one less than the number of its vertices. 4 Trees 39 Proof. We use induction on m = |V |, the number of vertices of G. ) The case m = 1 is trivial. 31. 7) is acyclic. This is clearly true when it has one or two vertices. For the inductive step, suppose that it is true for any connected graph with m − 1 vertices and let G be a connected graph with m vertices.