Xuding Zhu National Sun Yat-sen University 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? Bipartite density Maximum-cut Problem Application in VLSI

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

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

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

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

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 . . .

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

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

Yu and Xu strict with 7 exceptions

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?

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?

Theorem A Theorem B Inequality strict with 7 exceptions

Are these numbers correct?

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

A reasonable guess:

?

? [Z,2008] with exceptions:

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

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

Theorem [Z, 2008]

Thank you !