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coloring_nested_tire_dual_graphs: tighten abstract/intro for moved Def

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The previous abstract/intro still treated "partial tire dual" as
foundational vocabulary defined elsewhere.  After moving Definition
1.7 into this paper, the wording is fixed:
  - Abstract: now lists tire graphs + dual depth as foundational
    (from companion paper), and notes we DEFINE partial tire dual
    here.
  - Intro: removes "partial tire duals D(T)" from the list of
    foundational vocabulary cited from the companion paper.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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didericis
2026-05-27 00:58:02 -04:00
parent df9411d2bd
commit 70b605e84f
3 changed files with 11 additions and 11 deletions
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papers/coloring_nested_tire_dual_graphs/paper.log
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Output written on paper.pdf (9 pages, 605744 bytes).
Output written on paper.pdf (9 pages, 605784 bytes).
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papers/coloring_nested_tire_dual_graphs/paper.pdf
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papers/coloring_nested_tire_dual_graphs/paper.tex
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\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 $G'$