1. Xuding Zhu National Sun Yat-sen University
Bipartite subgraphs of
subcubic triangle-free
graphs
Xuding Zhu
National Sun Yat-sen University

2. 2007年6月

3. Bipartite subgraphs
in subcubic graphs
2007年6月

4. 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?
Bipartite density
Maximum-cut Problem
Application in VLSI

5. Theorem
[Hopkins and Staton, 1982]
This bound is tight.

6. Theorem
[Hopkins and Staton, 1982]
[Bondy-Locke, 1986]
cubic
>
with two exceptions: The Petersen graph
and the dodecahedron

7. Theorem
[Hopkins and Staton, 1982]
[Bondy-Locke, 1986]
cubic
>
with two exceptions: The Petersen graph
and the dodecahedron
necessary

8. Journal of Combinatorial Theory, Series B 98 (2008) 516–537
Conjecture
Theorem [Xu-Yu, 2008]
Theorem
[Hopkins and Staton, 1982]
[Bondy-Locke, 1986]
G: subcubic
cubic
>
with two exceptions: The Petersen graph
and the dodecahedron 7
. . .
Journal of Combinatorial Theory, Series B 98 (2008) 516–537

9. Conjecture
Theorem [Xu-Yu, 2008]
Theorem
[Hopkins and Staton, 1982]
[Bondy-Locke, 1986]
G: subcubic
cubic
>
with two exceptions: The Petersen graph
and the dodecahedron 7
. . .

10. If one can obtain an induced bipartite subgraph
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

11. To obtain a bipartite subgraph, it suffices to delete
less than edges
edges
with 7 exceptions

12. Yu and Xu
strict with 7 exceptions

13. Theorem A
Theorem B
Inequality strict with 7 exceptions
Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?

14. Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?

15. Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?

16. Deleting which three vertices
gives a bipartite subgraph?
Deleting which three edges
gives a bipartite subgraph?

17. Theorem A
Theorem B
Inequality strict with 7 exceptions

19. Are these numbers correct?

21. Theorem [Fajtlowicz (1978), Staton (1979)]
Griggs and Murphy (1996)
Jones, 1990, a shorter proof
Heckman and Thomas (2001) ]
Sharp!
A linear time algorithm finding an independent set
of size

22. A reasonable guess:

24. ?

25. ?
[Z,2008]
with exceptions:

26. Conjecture
Theorem [Xu-Yu, 2008]
Theorem [ Bondy-Locke (1986)]
[ Z, ]
G: subcubic
with two exceptions: 7

29. There is a multi-set of independent sets of average
size 5n/14, that `evenly’ covers the vertices of G

31. Theorem [Z, 2008]

35. Thank you !