ACO Seminar 2008-05-02

Title:Packing in Multipartite Graphs
Speaker: Ryan Martin, Iowa State
When: May 2, 11:30-12:30
Where: Hamburg Hall, Room 237


We present some results on packing graphs in dense multipartite graphs. This is a question very similar to the Hajnal-Szemeredi theorem, which gives sufficient minimum-degree conditions for an $n$-vertex graph to have a subgraph consisting of $\lfloor n/r\rfloor$ vertex-disjoint copies of $K_r$. This is a packing, or tiling, of the graph by copies of $K_r$. The Hajnal-Szemeredi theorem has been generalized to finding minimum-degree conditions that guarantee packings of non-complete graphs, notably by Alon and Yuster and by Kuhn and Osthus. We consider a multipartite version of this problem. That is, given an $r$-partite graph with $N$ vertices in each partition, what is the minimum-degree required of the bipartite graph induced by each pair of color-classes so that the graph contains $N$ vertex-disjoint copies of $K_r$? The question has been answered for $r=3,4$, provided $r$ is sufficiently large. When $r=3$ and $N$ is sufficiently large, a degree condition of $(2/3)N$ is sufficient with the exception of a single tripartite graph when $N$ is an odd multiple of $3$. When $r=4$ and $N$ is sufficiently large, a degree condition of $(3/4)N$ is sufficient and there is no exceptional graph. There are also bounds on the degree condition for higher $r$ by Csaba and Mydlarz. This question has also been generalized to finding minimum-degree conditions for packings of some arbitrary $r$-colorable graph in an $r$-partite. The case $r=2$ is highly nontrivial for packing arbitrary bipartite graphs and was answered very precisely by Zhao. The case $r=3$ is even more complex and we provide some tight bounds on the required degree condition. This talk includes joint work with Cs. Magyar, with E. Szemeredi and with Y. Zhao.

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