Induced 2-Degenerate Subgraphs of Triangle-Free Planar Graphs
A graph is $k$-degenerate if every subgraph has minimum degree at most $k$. We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph. We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $\alpha_2(G)$. We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $\alpha_2(G) \geq \frac{6n - m - 1}{5}$. By Euler's Formula, this implies $\alpha_2(G) \geq \frac{4}{5}n$. We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most three, then $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.
2021 ◽
Vol vol. 23, no. 3
(Graph Theory)
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Keyword(s):
Keyword(s):
New Upper Bounds for the Heights of Some Light Subgraphs in 1-Planar Graphs with High Minimum Degree
2011 ◽
Vol Vol. 13 no. 3
(Combinatorics)
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Keyword(s):
1996 ◽
Vol 05
(06)
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pp. 877-883
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2020 ◽
Vol 12
(03)
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pp. 2050034
2020 ◽
Vol 12
(04)
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pp. 2050035
Keyword(s):
2011 ◽
Vol 50-51
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pp. 245-248