Decomposing planar graphs into graphs with degree restrictions

초록

Given a graph G $G$, a decomposition of G $G$ is a partition of its edges. A graph is ( d , h ) $(d,h)$-decomposable if its edge set can be partitioned into a d $d$-degenerate graph and a graph with maximum degree at most h $h$. For d <= 4 $d\le 4$, we are interested in the minimum integer h d $h_d$ such that every planar graph is ( d , h d ) $(d,h_d)$-decomposable. It was known that h 3 <= 4 $h_3\le 4$, h 2 <= 8 $h_2\le 8$, and h 1 = infinity $h_1=\infty$. This paper proves that h 4 = 1 , h 3 = 2 $h_4=1,h_3=2$, and 4 <= h 2 <= 6 $4\le h_2\le 6$.

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