Book (graph theory)

In graph theory, a book graph (often written $B_p$ ) 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] A book of this type is the Cartesian product of a star and K₂ . 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 K_1,1,p. It is a graph consisting of $p$ triangles sharing a common edge.^[3] A book of this type is a split graph. This graph has also been called a $K_e(2,p)$.^[4]

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

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 $B_p$ for his book-graph.)

Theorems on books

Denote the Ramsey number of two (triangular) books by $r(B_p,\ B_q).$

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

References

1. Weisstein, Eric W., "Book Graph", MathWorld.
2. 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 K_2,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.