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Light subgraphs in planar graphs of minimum degree 4 and edge-degree 9. (English) Zbl 1031.05075
Summary: A graph $$H$$ is light in a given class of graphs if there is a constant $$w$$ such that every graph of the class which has a subgraph isomorphic to $$H$$ also has a subgraph isomorphic to $$H$$ whose sum of degrees in $$G$$ is $$\leq w$$. Let $$\mathcal G$$ be the class of simple planar graphs of minimum degree $$\geq 4$$ in which no two vertices of degree 4 are adjacent. We denote the minimum such $$w$$ by $$w(H)$$. It is proved that the cycle $$C_s$$ is light if and only if $$3 \leq s \leq 6$$, where $$w(C_3) = 21$$ and $$w(C_4)\leq 35$$. The 4-cycle with one diagonal is not light in $$\mathcal G$$, but it is light in the subclass $$\mathcal G^T$$ consisting of all triangulations. The star $$K_{1,s}$$ is light if and only if $$s\leq 4$$. In particular, $$w(K_{1,3}) = 23$$. The paths $$P_s$$ are light for $$1 \leq s \leq 6$$, and heavy for $$s\geq 8$$. Moreover, $$w(P_3) = 17$$ and $$w(P_4) = 23$$.

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