coloring_nested_tire_graphs: fix menagerie -- θ(1,p,q) IS outerplanar (cycle + chord)
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 <noreply@anthropic.com>
This commit is contained in:
@@ -1,2 +1,3 @@
|
||||
\relax
|
||||
\@writefile{toc}{\contentsline {paragraph}{Closed form.}{3}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{4}
|
||||
|
||||
@@ -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.
|
||||
<fig_blocktree.png, id=29, 242.55618pt x 147.70181pt>
|
||||
File: fig_blocktree.png Graphic file (type png)
|
||||
<use fig_blocktree.png>
|
||||
Package pdftex.def Info: fig_blocktree.png used on input line 122.
|
||||
Package pdftex.def Info: fig_blocktree.png used on input line 158.
|
||||
(pdftex.def) Requested size: 258.36668pt x 157.34296pt.
|
||||
<fig_theta.png, id=30, 168.47943pt x 147.70181pt>
|
||||
[3]
|
||||
<fig_theta.png, id=38, 168.47943pt x 147.70181pt>
|
||||
File: fig_theta.png Graphic file (type png)
|
||||
<use fig_theta.png>
|
||||
Package pdftex.def Info: fig_theta.png used on input line 146.
|
||||
Package pdftex.def Info: fig_theta.png used on input line 183.
|
||||
(pdftex.def) Requested size: 150.32504pt x 131.79225pt.
|
||||
[3 <./fig_blocktree.png>] [4 <./fig_theta.png>] (./menagerie.aux) )
|
||||
[4 <./fig_blocktree.png> <./fig_theta.png>] (./menagerie.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
4626 strings out of 478268
|
||||
75252 string characters out of 5846347
|
||||
370157 words of memory out of 5000000
|
||||
22807 multiletter control sequences out of 15000+600000
|
||||
478664 words of font info for 64 fonts, out of 8000000 for 9000
|
||||
4643 strings out of 478268
|
||||
75478 string characters out of 5846347
|
||||
373157 words of memory out of 5000000
|
||||
22821 multiletter control sequences out of 15000+600000
|
||||
481374 words of font info for 75 fonts, out of 8000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
55i,6n,63p,245b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/local
|
||||
/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/local/
|
||||
texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/t
|
||||
exlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/te
|
||||
xlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></usr/local/tex
|
||||
live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texli
|
||||
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive
|
||||
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/local/texlive/2
|
||||
022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/local/texlive/202
|
||||
2/texmf-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/t
|
||||
exmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texm
|
||||
f-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf
|
||||
-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-d
|
||||
ist/fonts/type1/public/amsfonts/cm/cmti10.pfb>
|
||||
Output written on menagerie.pdf (4 pages, 197957 bytes).
|
||||
{/usr/local/tex
|
||||
live/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc}</usr/local/texl
|
||||
ive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texli
|
||||
ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/local/texliv
|
||||
e/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive
|
||||
/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/
|
||||
2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></usr/local/texlive/2
|
||||
022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/local/texlive/202
|
||||
2/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/
|
||||
texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/local/texlive/2022/te
|
||||
xmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/local/texlive/2022/texm
|
||||
f-dist/fonts/type1/public/amsfonts/cm/cmr6.pfb></usr/local/texlive/2022/texmf-d
|
||||
ist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist
|
||||
/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/
|
||||
fonts/type1/public/amsfonts/cm/cmsy8.pfb></usr/local/texlive/2022/texmf-dist/fo
|
||||
nts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fon
|
||||
ts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/font
|
||||
s/type1/public/cm-super/sfrm1095.pfb>
|
||||
Output written on menagerie.pdf (4 pages, 236404 bytes).
|
||||
PDF statistics:
|
||||
95 PDF objects out of 1000 (max. 8388607)
|
||||
50 compressed objects within 1 object stream
|
||||
111 PDF objects out of 1000 (max. 8388607)
|
||||
60 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
31 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
@@ -101,20 +101,56 @@ processed edges incident to the new endpoint of $e$ (between $1$ and $2$, since
|
||||
$\Delta\le 3$). In practice this gives a clean product depending only on the
|
||||
degree sequence of $T$.
|
||||
|
||||
\subsection*{6.~Two-connected outerplanar with $\Delta=3$ is just $C_n$}
|
||||
\subsection*{6.~Two-connected outerplanar with $\Delta\le 3$: cycle, possibly with a matching of chords}
|
||||
|
||||
The only $2$-connected outerplanar graphs are polygons (with optional chords).
|
||||
Each chord adds a degree to each of its two endpoints; if every vertex on the
|
||||
polygon $C_n$ already has degree $2$ from the cycle, then we may add at most
|
||||
one chord-endpoint per vertex, so chords must form a matching. Already a
|
||||
\emph{single} chord on a $2$-connected outerplanar graph forces both
|
||||
endpoints to degree $3$. In particular, the only $2$-connected outerplanar
|
||||
graph with $\Delta\le 3$ in which the maximum is actually attained at
|
||||
\emph{every} vertex would be a polygon with a perfect matching of chords;
|
||||
but each chord crosses some other (mod the planar embedding) unless the two
|
||||
matched vertices are adjacent on the polygon, which collapses the
|
||||
``$2$-connected'' assumption. The upshot is: \emph{the $2$-connected blocks
|
||||
in our class are just cycles.}
|
||||
The $2$-connected outerplanar graphs are polygons (with optional chords).
|
||||
Each chord raises the degree of its two endpoints by $1$, and a polygon
|
||||
vertex already has degree $2$ from the cycle. So for $\Delta\le 3$ the
|
||||
chords must form a \emph{matching} of polygon vertices --- no vertex
|
||||
can be an endpoint of two chords. In particular the simplest non-cycle
|
||||
case is a polygon with a \emph{single} chord, denoted $\theta(1,p,q)$:
|
||||
two paths of lengths $p$ and $q$ between two trivalent vertices, plus a
|
||||
direct edge between those two trivalent vertices. Equivalently it is
|
||||
$C_{p+q}$ with a chord joining the two cycle vertices at distance $p$
|
||||
apart on the polygon.
|
||||
|
||||
\begin{center}
|
||||
\textit{[$\theta(1,p,q)$: polygon $C_{p+q}$ with one chord; both chord
|
||||
endpoints have degree~$3$, other polygon vertices have degree~$2$.]}
|
||||
\end{center}
|
||||
|
||||
$\theta(1,p,q)$ is outerplanar (the chord lies inside the polygon, and
|
||||
all polygon vertices are on the outer face) and subcubic.
|
||||
|
||||
\paragraph{Closed form.}
|
||||
For each choice of chord color $c\in\{1,2,3\}$, the two cycle edges
|
||||
incident to each chord endpoint must lie in $\{x,y\} = \{1,2,3\}\setminus\{c\}$.
|
||||
Conditioning on whether the two pairs of cycle-edge colors at the chord
|
||||
endpoints agree or disagree, and using the transfer matrix
|
||||
$T = J - I$ on $K_3$ (eigenvalues $2$ and~$-1$ with multiplicities $1$ and $2$),
|
||||
one finds
|
||||
\[
|
||||
\boxed{\;P_e(\theta(1,p,q),\,3)
|
||||
\;=\; \dfrac{2^{p+q} - 2^{p}(-1)^{q} - 2^{q}(-1)^{p} + 10\,(-1)^{p+q}}{3}\;}.
|
||||
\]
|
||||
Sanity checks:
|
||||
\begin{itemize}
|
||||
\item $p = q = 2$: $\theta(1,2,2) = K_4 - e$, formula gives
|
||||
$(16 - 4 - 4 + 10)/3 = 6 = P_e(K_4 - e, 3)$.
|
||||
\item $p = q = 3$: $\theta(1,3,3)$, formula gives
|
||||
$(64 + 8 + 8 + 10)/3 = 30$.
|
||||
\item $p = q = 6$: $\theta(1,6,6)$ (= the interior dual subgraph of
|
||||
the partial tire dual for the barbell-$O$ tire of Figure~4 of
|
||||
the main paper), formula gives $(4096 - 64 - 64 + 10)/3 = 1326$.
|
||||
\end{itemize}
|
||||
The formula has been verified empirically against \texttt{Sage}'s
|
||||
chromatic polynomial routine for all
|
||||
$p, q \in \{2, 3, 4, 5, 6\}$.
|
||||
|
||||
More generally, a polygon with $r$ chords forming a matching has
|
||||
chromatic polynomial computable by the same transfer-matrix idea
|
||||
along the $r{+}1$ paths between consecutive chord endpoints on the
|
||||
polygon, with a product constraint at each chord endpoint.
|
||||
|
||||
\subsection*{7.~Block--cut decomposition}
|
||||
|
||||
@@ -140,20 +176,23 @@ cut-vertex constraint. For each cycle-block $B = C_n$ contributing
|
||||
$2^n + 2(-1)^n$ proper $3$-edge-colorings, and each edge-block contributing
|
||||
$3$, this product is computable in time linear in $|V(G)|+|E(G)|$.
|
||||
|
||||
\section*{Outside the menagerie: theta graphs}
|
||||
\section*{Outside the menagerie: $\theta(p,q,r)$ with all paths
|
||||
$\ge 2$, i.e.\ $K_{2,3}$ subdivisions}
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.32\textwidth]{fig_theta.png}
|
||||
\end{center}
|
||||
|
||||
The complete bipartite graph $K_{2,3}$, equivalently the theta graph
|
||||
$\theta(2,2,2)$, is \emph{not} outerplanar (it is a forbidden minor for
|
||||
outerplanarity). In our tire-graph application this is the structure of the
|
||||
interior dual subgraph of $D(T)$ when the inner outerplanar graph $O$ has a
|
||||
bridge: two trivalent vertices $d_f$ connected by three internally
|
||||
vertex-disjoint paths. Such a $D(T)$ falls outside the simple block-cut
|
||||
menagerie above and its $P_e(\cdot,3)$ does not reduce to a product over
|
||||
cycle-blocks; instead it is computed directly by deletion--contraction on the
|
||||
theta-graph structure, or via a transfer matrix on the three paths.
|
||||
The genuine theta graph $\theta(2,2,2) = K_{2,3}$ (and more generally any
|
||||
$\theta(p,q,r)$ with $p,q,r\ge 2$) is \emph{not} outerplanar: $K_{2,3}$
|
||||
is a forbidden minor for outerplanarity, and every $\theta(p,q,r)$ with
|
||||
$\min(p,q,r)\ge 2$ contains $K_{2,3}$ as a topological minor. Such graphs
|
||||
do \emph{not} arise as the interior dual subgraph of a partial tire dual
|
||||
$D(T)$ in this paper: when the inner outerplanar graph $O$ has a bridge,
|
||||
the interior dual subgraph is the polygon-with-one-chord case
|
||||
$\theta(1, p, q)$ of \S6, where the path of length~$1$ is precisely the
|
||||
dual of the bridge. In particular $D(T)$ \emph{is} subcubic outerplanar
|
||||
in the bridge case, and its edge $3$-coloring count is the
|
||||
$\theta(1,p,q)$ closed form above.
|
||||
|
||||
\end{document}
|
||||
|
||||
Reference in New Issue
Block a user