Generalized Lucas cubes

Let f be is a binary string and d?1. Then the generalized Lucas cube Qd(f?)is introduced as the graph obtained from the Qd by removing all vertices that have a circulation containing f as a substring. The question for which f and d, the generalized Lucas cube Qd(f?) is an isometric subgraph of the d-cube Qd is solved for all binary strings of length at most five. Several isometrically embeddable and non-embeddable infinite series where f is of arbitrary length are given. Some structural properties of generalized Lucas cubes are also presented.

Hyperbolicity and Chordality of a Graph

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.