SIAM STUDENT CHAPTER
UNIVERSITY OF SOUTH CAROLINA
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SIAM Student Chapter Seminar "Sparse Graphs are Note Flammable" Dr. Pawel Pralat Ryerson University, Toronto
Time: Tuesday, Nov. 15, 5:00 p.m. Location: LeConte 312 Abstract: We consider the following $k$-firefighter problem on a finite graph $G=(V,E)$. Suppose that a fire breaks out at a given vertex $v$ in $V$. In each subsequent time unit, a firefighter protects $k$ vertices which are not yet on fire, and then the fire spreads to all unprotected neighbours of the vertices on fire. The objective of the firefighter is to save as many vertices as possible. The surviving rate $\rho(G)$ of
$G$ is defined as the expected percentage of vertices that can be saved
when a fire breaks out at a random vertex of $G$. Let
$\tau_k=k+2-\frac{1}{k+2}$. We show that for any $\epsilon>0$
and $k \geq 2$, each graph $G$ on $n$ vertices with at most
$(\tau_k-\epsilon)n$ edges is not flammable; that is,
$rho(G)>\frac{2\epsilon}{5\tau_k}>0$. Moreover, a
construction of a family of flammable random graphs is proposed to show
that the constant $\tau_k$ cannot be improved. Surprisingly, for
$k=1$, the threshold is $\frac{30}{11}$, not $\tau_1=\frac{8}{3}$. |
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