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The 7-cycle $$C_{7}$$ is light in the family of planar graphs with minimum degree 5. (English) Zbl 1115.05022
Summary: A connected graph $$H$$ is said to be light in the family of graphs $${\mathcal H}$$ there exists a positive integer $$k$$ such that each graph $$G\in {\mathcal H}$$ that contains an isomorphic copy of $$H$$ contains a subgraph $$K$$ isomorphic to $$H$$ that satisfies the inequality $\sum_{v\in V(K)} \deg_G(v)\leq k.$ It is known that an $$r$$-cycle $$C_r$$ is light in the family of planar graphs with minimum degree 5 if $$3\leq r\leq 6$$, and not light for $$r\geq 11$$. We prove that $$C_7$$ is also light in this family.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles
##### Keywords:
planar graphs; light graphs; light cycles
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##### References:
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