graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, a book graph (often written

B_p

) may be any of several kinds of graph formed by multiple cycles sharing an edge.

Variations

One kind, which may be called a quadrilateral book, consists of ''p''

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

s sharing a common edge (known as the "spine" or "base" of the book). That is, it is a

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

of a

star A star is a luminous spheroid of plasma (physics), plasma held together by Self-gravitation, self-gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night sk ...

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 ''K''_1,1,''p''. It is a graph consisting of

p

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

s sharing a common edge. A book of this type is a split graph. This graph has also been called a

K_e(2,p)

or a thagomizer graph (after thagomizers, the spiked tails of

stegosauria Stegosauria is a group of Herbivore, herbivorous ornithischian dinosaurs that lived during the Jurassic and early Cretaceous Period (geology), periods. Stegosaurian fossils have been found mostly in the Northern Hemisphere (North America, Europe a ...

n dinosaurs, because of their pointy appearance in certain drawings) and their graphic matroids have been called thagomizer matroids. Triangular books form one of the key building blocks of line perfect graphs. 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

B_p

for his book-graph.)

Within larger graphs

Given a graph

G

, one may write

bk(G)

for the largest book (of the kind being considered) contained within

G

Theorems on books

Denote the Ramsey number of two triangular books by

r(B_p,\ B_q).

This is the smallest number

r

such that for every

r

-vertex graph, either the graph itself contains

B_p

as a subgraph, or its

complement graph In the mathematical field of graph theory, the complement or inverse of a graph is a graph on the same vertices such that two distinct vertices of are adjacent if and only if they are not adjacent in . That is, to generate the complement of ...

contains

B_q

as a subgraph. * If

1\leq p\leq q

, then

r(B_p,\ B_q)=2q+3

. * 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

References

{{DEFAULTSORT:Book (Graph Theory) Parametric families of graphs Planar graphs