Book (graph theory)

A triangular book

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] 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] Triangular books form one of the key building blocks of line perfect graphs.^[5]

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^[6] 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) $B_{k}$ .^[7]

References

↑ Weisstein, Eric W. "Book Graph". MathWorld.
1 2 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 K_2,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.
↑ Maffray, Frédéric (1992). "Kernels in perfect line-graphs". Journal of Combinatorial Theory. Series B. 55 (1): 1–8. doi:10.1016/0095-8956(92)90028-V. MR 1159851. .
↑ 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.

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[1] Weisstein, Eric W. "Book Graph". MathWorld.

[gallian-2] 1 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] Maffray, Frédéric (1992). "Kernels in perfect line-graphs". Journal of Combinatorial Theory. Series B. 55 (1): 1–8. doi:10.1016/0095-8956(92)90028-V. MR 1159851. .

[6] 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

[7] P. Erdos, On a theorem of Rademacher-Turán. Illinois Journal of Mathematics, vol. 6 (1962), pp. 122–127.