## A criterion for the embeddability of $$(r,q)$$-polycycles.(English. Russian original)Zbl 1045.05035

Russ. Math. Surv. 57, No. 3, 589-591 (2002); translation from Usp. Mat. Nauk 57, No. 3, 149-150 (2002).
An $$(r,q)$$-polycycle is a plane graph all of whose inner faces are $$r$$-gons and all of whose inner vertices have degree $$q$$. A graph is said to be $$\lambda$$-embeddable if it can be embedded into a hypercube so that the combinatorial distances between vertices get multiplied by $$\lambda$$. The paper gives a necessary and sufficient condition for a $$(3,5)$$-polycycle to be $$\lambda$$-embeddable. Definitions and prerequisites can be found in a paper by M.-M. Deza and M. Shtogrin [J. Geom. Phys. 40, 302–319 (2002; Zbl 0996.05081)].

### MSC:

 05C10 Planar graphs; geometric and topological aspects of graph theory 05C12 Distance in graphs

Zbl 0996.05081
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