Book (graph theory)
In graph theory, a book graph (often written ) may be any of several kinds of graph.
One kind, which may be called a quadrilateral book, consists of p quadrilaterals sharing a common edge (known as the "spine" or "base" of the book). That is, it is a Cartesian product of a star and a single edge. The 7-page book graph of this type provides an example of a graph with no harmonious labeling.
A second type, which might be called a triangular book, is the complete tripartite graph K1,1,p. It is a graph consisting of triangles sharing a common edge. A book of this type is a split graph. This graph has also been called a .
Given a graph , one may write for the largest book (of the kind being considered) contained within .
The term "book-graph" has been employed for other uses. Barioli used it to mean a graph composed of a number of arbitrary subgraphs having two vertices in common. (Barioli did not write for his book-graph.)
Theorems on books
Denote the Ramsey number of two (triangular) books by
- If , then (proved by Rousseau and Sheehan).
- There exists a constant such that whenever .
- If , and is large, the Ramsey number is given by .
- Let be a constant, and . Then every graph on vertices and edges contains a (triangular) .
- Weisstein, Eric W. "Book Graph". MathWorld.
- Gallian, Joseph A. (1998), "A dynamic survey of graph labeling", Electronic Journal of Combinatorics, 5: Dynamic Survey 6, MR 1668059.
- Lingsheng Shi and Zhipeng Song, Upper bounds on the spectral radius of book-free and/or K2,l-free graphs. Linear Algebra and its Applications, vol. 420 (2007), pp. 526–529. doi:10.1016/j.laa.2006.08.007
- Erdős, Paul (1963). "On the structure of linear graphs". Israel Journal of Mathematics. 1: 156–160. doi:10.1007/BF02759702.
- Francesco Barioli, Completely positive matrices with a book-graph. Linear Algebra and its Applications, vol. 277 (1998), pp. 11–31. doi:10.1016/S0024-3795(97)10070-2
- P. Erdos, On a theorem of Rademacher-Turán. Illinois Journal of Mathematics, vol. 6 (1962), pp. 122–127.