diff --git a/papers/coloring_nested_tire_dual_graphs/paper.log b/papers/coloring_nested_tire_dual_graphs/paper.log index bd2c1c9..4d55b1f 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.log +++ b/papers/coloring_nested_tire_dual_graphs/paper.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) 27 MAY 2026 00:57 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:58 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -258,7 +258,7 @@ live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb> -Output written on paper.pdf (9 pages, 605744 bytes). +Output written on paper.pdf (9 pages, 605784 bytes). PDF statistics: 151 PDF objects out of 1000 (max. 8388607) 89 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_dual_graphs/paper.pdf b/papers/coloring_nested_tire_dual_graphs/paper.pdf index 607c03a..ad28826 100644 Binary files a/papers/coloring_nested_tire_dual_graphs/paper.pdf and b/papers/coloring_nested_tire_dual_graphs/paper.pdf differ diff --git a/papers/coloring_nested_tire_dual_graphs/paper.tex b/papers/coloring_nested_tire_dual_graphs/paper.tex index 9f8ce8f..97746a8 100644 --- a/papers/coloring_nested_tire_dual_graphs/paper.tex +++ b/papers/coloring_nested_tire_dual_graphs/paper.tex @@ -45,10 +45,11 @@ \begin{abstract} This is a follow-up to \cite{bauerfeld-nested-tires}, which -establishes the basic vocabulary of tire graphs $T$ and their -partial tire duals $D(T)$. Building on those definitions, we -analyse the structure of $D(T)$ in the spoke-only case (a corona -graph $C_{n+m} \circ K_1$), prove the tire-component lemma +establishes the basic vocabulary of tire graphs $T$ and dual +depth. Building on those definitions, we define the +\emph{partial tire dual} $D(T)$ and analyse its structure in the +spoke-only case (a corona graph $C_{n+m} \circ K_1$), prove the +tire-component lemma exhibiting every BFS-level component as a tire graph, give an edge-vertex coloring bijection that reduces counting proper $3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings @@ -72,11 +73,10 @@ admitting no proper $3$-edge-colouring. This paper is the second in a series studying that structure through the lens of \emph{nested level duals}. The foundational vocabulary --- level sources, levels, the inner planar dual $G'$ -and its dual depth, tire graphs, and partial tire duals -$D(T)$ --- is developed in the companion paper -\cite{bauerfeld-nested-tires}; we refer to that paper for all -basic definitions and rely on them throughout. In particular we -use, without restating, the notions of: +and its dual depth, and tire graphs --- is developed in the +companion paper \cite{bauerfeld-nested-tires}; we refer to that +paper for those definitions and rely on them throughout. In +particular we use, without restating, the notions of: \begin{itemize} \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$; \item the inner planar dual $G'$