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DTSTART;TZID=Asia/Seoul:20201105T100000
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SUMMARY:Daniel Cranston\, Vertex Partitions into an Independent Set and a Forest with Each Component Small
DESCRIPTION:For each integer $k\ge 2$\, we determine a sharp bound on\n$\operatorname{mad}(G)$ such that $V(G)$ can be partitioned into sets $I$ and $F_k$\, where $I$ is an independent set and $G[F_k]$ is a forest in which each component has at most k vertices. For each $k$ we construct an infinite family of examples showing our result is best possible. Hendrey\, Norin\, and Wood asked for the largest function $g(a\,b)$ such that if $\operatorname{mad}(G) < g(a\,b)$ then $V(G)$ has a partition into sets $A$ and $B$ such that $\operatorname{mad}(G[A]) < a$ and $\operatorname{mad}(G[B]) < b$. They specifically asked for the value of $g(1\,b)$\, which corresponds to the case that $A$ is an independent set. Previously\, the only values known were $g(1\,4/3)$ and $g(1\,2)$. We find the value of $g(1\,b)$ whenever $4/3 < b < 2$. This is joint work with Matthew Yancey.
URL:https://dimag.ibs.re.kr/event/2020-11-05/
LOCATION:Zoom ID: 869 4632 6610 (ibsdimag)
CATEGORIES:Virtual Discrete Math Colloquium
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