From e47f89918acd6d346dcf8b61762f790cd9b4562c Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 21:19:08 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20fix=20mena?= =?UTF-8?q?gerie=20--=20=CE=B8(1,p,q)=20IS=20outerplanar=20(cycle=20+=20ch?= =?UTF-8?q?ord)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit User correctly pointed out: (1) The Figure 4 partial-tire-dual interior structure is not a "theta graph" in the K_{2,3} sense (which requires all three paths of length ≥ 2). It is θ(1, 6, 6): a 12-cycle with one chord. (2) θ(1, p, q) IS outerplanar (just a polygon with one chord), so it belongs IN the menagerie, not outside it. Revisions: - Section 6 ("2-connected outerplanar with Δ ≤ 3"): previously claimed the class is just cycles; corrected to "cycle, possibly with a matching of chords." Added explicit description of θ(1, p, q) and a closed-form for its proper 3-edge-coloring count: P_e(θ(1,p,q), 3) = (2^{p+q} - 2^p (-1)^q - 2^q (-1)^p + 10 (-1)^{p+q}) / 3. Verified against Sage's chromatic polynomial for all p, q ∈ {2..6}. - "Outside the menagerie" section: previously said "theta graphs (all flavours) are not outerplanar." Corrected to clarify that only θ(p, q, r) with all three paths of length ≥ 2 (= K_{2,3} subdivisions) is not outerplanar. Explicitly noted that the bridge-case partial tire dual gives θ(1, p, q) which IS in the menagerie, with edge-3- coloring count given by the closed form. The Figure 4 partial-tire-dual (m=4 outer cycle + barbell O with bridge) has θ(1, 6, 6) as its interior dual subgraph and so admits exactly 1326 proper 3-edge-colorings on the interior cycle-with- chord; leaves contribute their forced colors as in the spoke-only case. Paper unchanged. This is a correction within the notes/ subdir only. Co-Authored-By: Claude Opus 4.7 --- .../notes/menagerie.aux | 1 + .../notes/menagerie.log | 59 ++++++------ .../notes/menagerie.pdf | Bin 197957 -> 236404 bytes .../notes/menagerie.tex | 85 +++++++++++++----- 4 files changed, 95 insertions(+), 50 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.aux b/papers/coloring_nested_tire_graphs/notes/menagerie.aux index 4089a83..5dd1744 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.aux +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.aux @@ -1,2 +1,3 @@ \relax +\@writefile{toc}{\contentsline {paragraph}{Closed form.}{3}{}\protected@file@percent } \gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/menagerie.log b/papers/coloring_nested_tire_graphs/notes/menagerie.log index 1a1cce1..98c8d48 100644 --- a/papers/coloring_nested_tire_graphs/notes/menagerie.log +++ b/papers/coloring_nested_tire_graphs/notes/menagerie.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:05 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 21:18 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -293,40 +293,45 @@ Package pdftex.def Info: fig_corona.png used on input line 76. 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