Bipartite subgraphs of
subcubic triangle-free
graphs
Xuding Zhu
National Sun Yat-sen University
2007年6月
Bipartite subgraphs
in subcubic graphs
2007年6月
A graph G is subcubic = maximum degree at most 3
Subcubic triangle free = subcubic + triangle free
What is the maximum number of edges in a bipartite
subgraph of a subcubic triangle free graph?
 2 (G)  max| E ( H ) |: H is a bi. subg. of G
Bipartite density
b(G )   2 (G ) / | E (G ) |
Maximum-cut Problem
Application in VLSI
Theorem [Hopkins and Staton, 1982]
G subcubic and triangle - free  b(G )  4
5
This bound is tight.
Theorem [Hopkins
[Bondy-Locke,
and Staton,
1986] 1982]
cubic and triangle - free  b(G ) 
G subcubic
>4
with two exceptions: The Petersen graph
and the dodecahedron
5
Theorem [Hopkins
[Bondy-Locke,
and Staton,
1986] 1982]
cubic and triangle - free  b(G ) 
G subcubic
>4
with two exceptions: The Petersen graph
and the dodecahedron
necessary
5
Theorem
[Hopkins
and Staton,
Conjecture
Theorem [Xu-Yu,
[Bondy-Locke,
2008]
1986] 1982]
G: subcubic
subcubic
cubic and triangle - free  b(G ) 
G
>4
with two
7 exceptions: The Petersen graph
and the dodecahedron
Journal of Combinatorial Theory,
Series B 98 (2008) 516–537
5
...
Theorem
[Hopkins
and Staton,
1982]graph by deleting
Conjecture
Theorem
[Xu-Yu,
[Bondy-Locke,
1986]
Theorem
: One
can2008]
obtain
a bipartite
G: subcubic
subcubic
cubic
G
and |triangle
- free  b(G ) 
less than
E | / 5 edges,
>4
with two
7 exceptions: The Petersen graph
and the dodecahedron
5
...
For a subcubic graph,
If one can obtain an induced bipartite subgraph
by removing k vertices,
then one can obtain a bipartite subgraph
by removing k edges.
v
e
Instead of deleting v to make the graph bipartite
we can delete edge e to make it bipartite
To obtain a bipartite subgraph, it suffices to delete
less than | E | / 5 edges
ni : number of vertices of degree i
| E | (3n3  2n2  n1 ) / 2
Theorem : One can obtain a bipartite graph by deleting
edges
less than (3n3  2n2  n1 ) / 10 vertices,
with 7 exceptions
 2 (G ) : maximum number of vertices in an induced
bipartite subgraph of G
Theorem : G triangle - free  subcubic 
 2 (G)  (7n3  8n2  9n1 ) / 10
Yu and Xu
strict with 7 exceptions
Theorem A
Theorem B
G subcubic and  - free G subcubic and  - free
7n3  8n2  9n1
 b(G )  4
  2 (G ) 
5
10
Inequality strict with 7 exceptions
Deleting which three edges
gives a bipartite subgraph?
Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?
Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?
Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?
Deleting which three vertices
gives a bipartite subgraph?
Theorem A
Theorem B
G subcubic and  - free G subcubic and  - free
7n3  8n2  9n1
 b(G )  4
  2 (G ) 
5
10
Inequality strict with 7 exceptions
Theorem : G triangle - free  subcubic 
 2 (G)  (7n3  8n2 109n1 ) / 10
Each verte x makes some contributi on to  2 (G)
7
Each 3 - vertex contribute s to  2 (G )
10
8
Each 2 - vertex contribute s
to  2 (G )
10
10
9
Each 1 - vertex contribute s
to  2 (G )
10
Theorem : G triangle - free  subcubic 
 2 (G)  (7n3  8n2  10n1 ) / 10
(7, 8,
10)
Are these numbers correct?
7
Each 3 - vertex contribute s to  2 (G )
10
8
Each 2 - vertex contribute s
to  2 (G )
10
10
9
Each 1 - vertex contribute s
to  2 (G )
10
Theorem : G triangle - free  subcubic 
 2 (G)  (7n3  8n2 109n1 ) / 10
What is the correct contributi on of each verte x to  2 (G)?
7
Each 3 - vertex contribute s to  2 (G )
10
8
Each 2 - vertex contribute s
to  2 (G )
10
10
9
Each 1 - vertex contribute s
to  2 (G )
10
Theorem [Fajtlowicz
Jones, and
1990,
(1978),
aThomas
shorter
Staton
proof
(1979)]
Griggs
Murphy
(1996)
Sharp!
Heckman
and
(2001)
]
 (G )  5 | V (G ) | / 14
 (G ) : independen ce number of G
 (G)  5.
A linear time algorithm finding an independent set
of size 5(n  k ) / 14
5n / 14
 (G )  5 | V (G ) | / 14
5
Each 3 - vertex contribute s to  (G)
14
An induced bipartite subgraph is the union of
2 independen t sets.
A reasonable guess:
5
5
Each 3 - vertex contribute s  2  to  2 (G )
14
7
Theorem : G triangle - free  subcubic 
 2 (G)  (7n3  8n2 109n1 ) / 10
What is the correct contributi on of each verte x to  2 (G)?
5
7
Each 3 - vertex contribute s to  2 (G )
7
10
?8
Each 2 - vertex contribute s
to  2 (G )
10
7
10
79
Each 1 - vertex contribute s
to  2 (G )
10
7
Theorem ?: G triangle - free  subcubic 
7 n1 ) / 10
 2 (G)  ( 57n3  68n2  10
7
What is the correct contributi on of each verte x to  2 (G)?
5
7
Each 3 - vertex contribute s to  2 (G )
7
10
6
?8
Each 2 - vertex contribute s
to  2 (G )
10
7
10
79
Each 1 - vertex contribute s
to  2 (G )
10
7
Theorem ?: G triangle - free  subcubic 
7 n1 ) / 10
[Z,2008]  2 (G)  ( 57 n3  68n2  10
7 with exceptions:
Theorem
2008]] (1986)]
Conjecture
Theorem [Xu-Yu,
[[ Bondy-Locke
Z, 2008
17
4

G
cubic
and
triangle
free

b
(
G
)

G: subcubic
521
with two
7 exceptions:
Theorem
G : subcubic and triangle - free 
 2 (G )  5 | V (G ) | / 7, with 2 exceptions .
Theorem
G : subcubic and triangle - free 
 2 (G )  5 | V (G ) | / 7, with 2 exceptions .
The re are 2 disjoing independen t sets of average
5
size
| V(G) | .
14
Conjecture [Heckman - Thomas]
If G is subcubuc triangle - free,
14
1
then  f (G ) 
 3
5
5
There is a multi-set of independent sets of average
size 5n/14, that `evenly’ covers the vertices of G
Theorem [Hatami - Z, ]
3
 f (G )  3 
64
Theorem [Z, 2008]
 2 (G)   (G)   (G)
5n3  6n2  7 n1
 (G ) 
7
-2
Graphs G with  (G) 
7
-1
Graphs G with  (G) 
7
Graphs G with  (G)  0
Thank you !