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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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papers/coloring_nested_tire_dual_graphs/paper.tex
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\begin{abstract} \begin{abstract}
This is a follow-up to \cite{bauerfeld-nested-tires}, which This is a follow-up to \cite{bauerfeld-nested-tires}, which
establishes the basic vocabulary of tire graphs $T$ and their establishes the basic vocabulary of tire graphs $T$ and dual
partial tire duals $D(T)$. Building on those definitions, we depth. Building on those definitions, we define the
analyse the structure of $D(T)$ in the spoke-only case (a corona \emph{partial tire dual} $D(T)$ and analyse its structure in the
graph $C_{n+m} \circ K_1$), prove the tire-component lemma 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 exhibiting every BFS-level component as a tire graph, give an
edge-vertex coloring bijection that reduces counting proper edge-vertex coloring bijection that reduces counting proper
$3$-edge-colorings of $D(T)$ to counting proper $3$-vertex-colorings $3$-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 This paper is the second in a series studying that structure
through the lens of \emph{nested level duals}. The foundational through the lens of \emph{nested level duals}. The foundational
vocabulary --- level sources, levels, the inner planar dual $G'$ vocabulary --- level sources, levels, the inner planar dual $G'$
and its dual depth, tire graphs, and partial tire duals and its dual depth, and tire graphs --- is developed in the
$D(T)$ --- is developed in the companion paper companion paper \cite{bauerfeld-nested-tires}; we refer to that
\cite{bauerfeld-nested-tires}; we refer to that paper for all paper for those definitions and rely on them throughout. In
basic definitions and rely on them throughout. In particular we particular we use, without restating, the notions of:
use, without restating, the notions of:
\begin{itemize} \begin{itemize}
\item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$; \item \emph{level source} $S$ and $G$-vertex levels $\ell_G(v)$;
\item the inner planar dual $G'$ \item the inner planar dual $G'$