- Source: Book (graph theory)
In graph theory, a book graph (often written
B
p
{\displaystyle 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 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. 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 K1,1,p. It is a graph consisting of
p
{\displaystyle p}
triangles sharing a common edge. A book of this type is a split graph.
This graph has also been called a
K
e
(
2
,
p
)
{\displaystyle K_{e}(2,p)}
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. 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
{\displaystyle B_{p}}
for his book-graph.)
Within larger graphs
Given a graph
G
{\displaystyle G}
, one may write
b
k
(
G
)
{\displaystyle bk(G)}
for the largest book (of the kind being considered) contained within
G
{\displaystyle G}
.
Theorems on books
Denote the Ramsey number of two triangular books by
r
(
B
p
,
B
q
)
.
{\displaystyle r(B_{p},\ B_{q}).}
This is the smallest number
r
{\displaystyle r}
such that for every
r
{\displaystyle r}
-vertex graph, either the graph itself contains
B
p
{\displaystyle B_{p}}
as a subgraph, or its complement graph contains
B
q
{\displaystyle B_{q}}
as a subgraph.
If
1
≤
p
≤
q
{\displaystyle 1\leq p\leq q}
, then
r
(
B
p
,
B
q
)
=
2
q
+
3
{\displaystyle r(B_{p},\ B_{q})=2q+3}
.
There exists a constant
c
=
o
(
1
)
{\displaystyle c=o(1)}
such that
r
(
B
p
,
B
q
)
=
2
q
+
3
{\displaystyle r(B_{p},\ B_{q})=2q+3}
whenever
q
≥
c
p
{\displaystyle q\geq cp}
.
If
p
≤
q
/
6
+
o
(
q
)
{\displaystyle p\leq q/6+o(q)}
, and
q
{\displaystyle q}
is large, the Ramsey number is given by
2
q
+
3
{\displaystyle 2q+3}
.
Let
C
{\displaystyle C}
be a constant, and
k
=
C
n
{\displaystyle k=Cn}
. Then every graph on
n
{\displaystyle n}
vertices and
m
{\displaystyle m}
edges contains a (triangular)
B
k
{\displaystyle B_{k}}
.
References
Kata Kunci Pencarian:
- Matematika diskrit
- Graf lengkap
- Penggalian data
- Daftar masalah matematika yang belum terpecahkan
- Bumi
- Teorema empat warna
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- Sambungan paket
- Springer Nature
- Book (graph theory)
- Spectral graph theory
- Glossary of graph theory
- Graph theory
- Topological graph theory
- Component (graph theory)
- Tree (graph theory)
- Graph minor
- Graph labeling
- Girth (graph theory)
