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Improved Bounds for Track Numbers of Planar Graphs

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Sergey Pupyrev

Discrete Mathematics · Submitted 30 Oct 2019

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We prove that every planar graph has track number at most $225$ and every planar $3$-tree has track number at most $25$.

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arXiv ID1910.14153 (version 2)
Pith IDUEGNJQLD, Pith's permanent record of this paper and its review
CitationSergey Pupyrev. "Improved Bounds for Track Numbers of Planar Graphs." arXiv:1910.14153 (2019). https://pith.science/paper/UEGNJQLD
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,899 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.99 · compares with prior results 0.68 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

A track layout of a graph consists of a vertex coloring and a total order of each color class, such that no two edges cross between any two color classes. The track number of a graph is the minimum number of colors required by a track layout of the graph. This paper improves lower and upper bounds on the track number of several families of planar graphs. We prove that every planar graph has track number at most $225$ and every planar $3$-tree has track number at most $25$. Then we show that there exist outerplanar graphs whose track number is $5$, which leads to the best known lower bound of $8$ for planar graphs. Finally, we investigate leveled planar graphs and tighten bounds on the track number of weakly leveled graphs, Halin graphs, and X-trees.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A track layout of a graph consists of a vertex coloring and a total order of each color class, such that no two edges cross between any two color classes. background 1.00 0.06 0.02 asserted 0.99
s1 The track number of a graph is the minimum number of colors required by a track layout of the graph. background 1.00 0.06 0.03 asserted 0.99
s2 This paper improves lower and upper bounds on the track number of several families of planar graphs. contribution 1.00 0.88 0.13 asserted 0.99
s3 We prove that every planar graph has track number at most $225$ and every planar $3$-tree has track number at most $25$. result 0.99 0.96 0.67 asserted 1.00
s4 Then we show that there exist outerplanar graphs whose track number is $5$, which leads to the best known lower bound of $8$ for planar graphs. result 0.98 0.90 0.09 asserted 1.00
s5 Finally, we investigate leveled planar graphs and tighten bounds on the track number of weakly leveled graphs, Halin graphs, and X-trees. contribution 0.96 0.94 0.06 asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A track layout of a graph consists of a vertex coloring and a total order of each color class, such that no two edges cross between any two color classes. The track number of a graph is the minimum number of colors required by a track layout of the graph. This paper improves lower and upper bounds on the track number of several families of planar graphs. We prove that every planar graph has track number at most $225$ and every planar $3$-tree has track number at most $25$. Then we show that there exist outerplanar graphs whose track number is $5$, which leads to the best known lower bound of $8$ for planar graphs. Finally, we investigate leveled planar graphs and tighten bounds on the track number of weakly leveled graphs, Halin graphs, and X-trees.

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