On light graphs in the family of 4-connected planar graphs
Discrete Mathematics
Light subgraphs of order at most 3 in large maps of minimum degree 5 on compact 2-manifolds
European Journal of Combinatorics - Special issue: Topological graph theory II
Every toroidal graph without adjacent triangles is (4, 1)*-choosable
Discrete Applied Mathematics
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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 ≤ w. Let $\cal G$ be the class of simple planar graphs of minimum degree ≥ 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 Cs is light if and only if 3 ≤ s ≤ 6, where w(C3) = 21 and w(C4) ≤ 35. The 4-cycle with one diagonal is not light in $\cal G$, but it is light in the subclass ${\cal G}^T$ consisting of all triangulations. The star K1,s is light if and only if s ≤ 4. In particular, w(K1,3) = 23. The paths Ps are light for 1 ≤ s ≤ 6, and heavy for s ≥ 8. Moreover, w(P3) = 17 and w(P4) = 23. © 2003 Wiley Periodicals, Inc. J Graph Theory 44: 261–295, 2003