Book (Graph Theory)

In graph theory, a book graph (often written ${\displaystyle B_p}$ ) may be any of several kinds of graph formed by multiple cycles sharing an edge.

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 ${\displaystyle p}$ 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] or a thagomizer graph (after thagomizers, the spiked tails of stegosaurian dinosaurs, because of their pointy appearance in certain drawings) and their graphic matroids have been called thagomizer matroids.^[5] Triangular books form one of the key building blocks of line perfect graphs.^[6]

The term "book-graph" has been employed for other uses. Barioli^[7] 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.)

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

Denote the Ramsey number of two triangular books by ${\displaystyle r(B_p,B_q).}$ This is the smallest number ${\displaystyle r}$ such that for every ${\displaystyle r}$-vertex graph, either the graph itself contains ${\displaystyle B_p}$ as a subgraph, or its complement graph contains ${\displaystyle B_q}$ as a subgraph.

• If ${\displaystyle 1\le p\le q}$, then ${\displaystyle r(B_p,B_q)=2q+3}$.^[8]
• There exists a constant ${\displaystyle c=o(1)}$ such that ${\displaystyle r(B_p,B_q)=2q+3}$ whenever ${\displaystyle q\ge cp}$.
• If ${\displaystyle p\le q/6+o(q)}$, 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) ${\displaystyle B_k}$.^[9]

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