# Book (graph theory)

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A triangular book

In graph theory, a book graph (often written ${\displaystyle B_{p}}$ ) 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.[1][2] The 7-page book graph of this type provides an example of a graph with no harmonious labeling.[2]

A second type, which might be called a triangular book, is the complete tripartite graph K1,1,p. It is a graph consisting of ${\displaystyle p}$ triangles sharing a common edge.[3] A book of this type is a split graph. This graph has also been called a ${\displaystyle K_{e}(2,p)}$.[4]

Given a graph ${\displaystyle G}$, one may write ${\displaystyle bk(G)}$ for the largest book (of the kind being considered) contained within ${\displaystyle G}$.

The term "book-graph" has been employed for other uses. Barioli[5] used it to mean a graph composed of a number of arbitrary subgraphs having two vertices in common. (Barioli did not write ${\displaystyle B_{p}}$ for his book-graph.)

## Theorems on books

Denote the Ramsey number of two (triangular) books by ${\displaystyle r(B_{p},\ B_{q}).}$

• If ${\displaystyle 1\leq p\leq q}$, then ${\displaystyle r(B_{p},\ B_{q})=2q+3}$ (proved by Rousseau and Sheehan).
• There exists a constant ${\displaystyle c=o(1)}$ such that ${\displaystyle r(B_{p},\ B_{q})=2q+3}$ whenever ${\displaystyle q\geq cp}$.
• If ${\displaystyle p\leq q/6+o(q)}$, and ${\displaystyle q}$ is large, the Ramsey number is given by ${\displaystyle 2q+3}$.
• Let ${\displaystyle C}$ be a constant, and ${\displaystyle k=Cn}$. Then every graph on ${\displaystyle n}$ vertices and ${\displaystyle m}$ edges contains a (triangular) ${\displaystyle B_{k}}$.[6]

## References

1. ^
2. ^ a b Gallian, Joseph A. (1998), "A dynamic survey of graph labeling", Electronic Journal of Combinatorics, 5: Dynamic Survey 6, MR 1668059.
3. ^ 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
4. ^ Erdős, Paul (1963). "On the structure of linear graphs". Israel Journal of Mathematics. 1: 156–160. doi:10.1007/BF02759702.
5. ^ 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
6. ^ P. Erdos, On a theorem of Rademacher-Turán. Illinois Journal of Mathematics, vol. 6 (1962), pp. 122–127.