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{\bf Yaokun Wu and Chengpeng Zhang}
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{\bf Hyperbolicity and Chordality of a Graph}
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Let $G$ be a connected graph with the usual shortest-path metric $d$.
The graph $G$ is $\delta$-hyperbolic provided for any vertices
$x,y,u,v$ in it, the two larger of the three sums
$d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ by at most
$2\delta.$ The graph $G$ is $k$-chordal provided it has no induced
cycle of length greater than $k.$ Brinkmann, Koolen and Moulton find
that every $3$-chordal graph is $1$-hyperbolic and that graph is not
$\frac{1}{2}$-hyperbolic if and only if it contains one of two special
graphs as an isometric subgraph. For every $k\geq 4,$ we show that a
$k$-chordal graph must be
$\frac{\lfloor\frac{k}{2}\rfloor}{2}$-hyperbolic and there does exist
a $k$-chordal graph which is not $\frac{\lfloor
\frac{k-2}{2}\rfloor}{2}$-hyperbolic. Moreover, we prove that a
$5$-chordal graph is $\frac{1}{2}$-hyperbolic if and only if it does
not contain any of a list of five special graphs as an isometric
subgraph.
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